The Annals of Probability2017 Vol 45 No 2 824ndash878DOI 10121415-AOP1076copy Institute of Mathematical Statistics 2017
MEAN-FIELD STOCHASTIC DIFFERENTIAL EQUATIONSAND ASSOCIATED PDES
BY RAINER BUCKDAHNlowastDagger JUAN LIdagger1SHIGE PENGDagger AND CATHERINE RAINERlowast
Universiteacute de Bretagne Occidentalelowast Shandong University Weihaidagger
and Shandong UniversityDagger
In this paper we consider a mean-field stochastic differential equa-tion also called the McKeanndashVlasov equation with initial data (t x) isin[0 T ] times R
d whose coefficients depend on both the solution Xtxs and its
law By considering square integrable random variables ξ as initial condi-tion for this equation we can easily show the flow property of the solution
Xtξs of this new equation Associating it with a process X
txPξs which co-
incides with Xtξs when one substitutes ξ for x but which has the advan-
tage to depend on ξ only through its law Pξ we characterize the function
V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] under appropriate regularity conditions
on the coefficients of the stochastic differential equation as the unique clas-sical solution of a nonlocal partial differential equation of mean-field typeinvolving the first- and the second-order derivatives of V with respect to itsspace variable and the probability law The proof bases heavily on a pre-liminary study of the first- and second-order derivatives of the solution ofthe mean-field stochastic differential equation with respect to the probabil-ity law and a corresponding Itocirc formula In our approach we use the notionof derivative with respect to a probability measure with finite second mo-ment introduced by Lions in [Cours au Collegravege de France Theacuteorie des jeu agravechamps moyens (2013)] and we extend it in a direct way to the second-orderderivatives
CONTENTS
1 Introduction 8252 Preliminaries 8283 The mean-field stochastic differential equation 8354 First-order derivatives of XtxPξ 8385 Second-order derivatives of XtxPξ 8536 Regularity of the value function 863
Received October 2014 revised October 20151Supported by the NSF of P R China (Nos 11071144 11171187 11222110) Shandong Province
(Nos BS2011SF010 JQ201202) Program for New Century Excellent Talents in University (NoNCET-12-0331) 111 Project (No B12023)
MSC2010 subject classifications Primary 60H10 secondary 60K35Key words and phrases Mean-field stochastic differential equation McKeanndashVlasov equation
value function PDE of mean-field type
824
MEAN-FIELD SDES AND ASSOCIATED PDES 825
7 Itocirc formula and PDE associated with mean-field SDE 870Acknowledgments 877References 877
1 Introduction Given a complete probability space (FP ) endowedwith a Brownian motion B = (Bt )tisin[0T ] and its filtration F = (Ft )tisin[0T ] aug-mented by all P -null sets and a sufficiently rich sub-σ -algebra F0 independentof B we consider the mean-field stochastic differential equation (SDE) alsoknown under the name of McKeanndashVlasov SDE
dXtxs = σ
(Xtx
s PXtxs
)dBs + b
(Xtx
s PXtxs
)ds
(11)s isin [t T ]Xtx
t = x isinRd
It is well known that under an appropriate Lipschitz assumption on the coeffi-cients this equation possesses for all (t x) isin [t T ]timesR
d a unique solution Xtxs s isin
[0 T ] For the classical SDE whose coefficients σ(xμ) = σ(x) b(xμ) = b(x)
depend only on x isin Rd but not on the probability measure μ it is well known
that the solution Xtxs 0 le t le s le T x isinR
d defines a flow and if the coefficientsare regular enough the unique classical solution of the partial differential equation(PDE)
parttV (t x) + 12 tr
(σσ lowast(x)D2
xV (t x)) + b(x)DxV (t x) = 0
(t x) isin [0 T ] timesRd(12)
V (T x) = (x) x isin Rd
is V (t x) = E[(XtxT )] (t x) isin [0 T ] times R
d But how about the above SDEwhose coefficients depend on (xμ) isin R
d times P2(Rd) where P2(R
d) denotes thespace of the probability measures over Rd with finite second moment Of coursefor an SDE with coefficients depending on (xμ) the solution Xtx
s 0 le s le t leT x isin R
d does obviously not define a flow But we see easily that if we replacethe deterministic initial condition X
txt = x isin R
d by a square integrable randomvariable X
tξt = ξ isin L2(Ft Rd)(= L2(Ft P Rd)) and consider the SDE
dXtξs = σ
(Xtξ
s PX
tξs
)dBs + b
(Xtξ
s PX
tξs
)ds
(13)s isin [t T ]Xtξ
t = ξ isin Rd
(where obviously in general Xtξ = Xtx |x=ξ ) then we have the flow property
For all 0 le t le s le T and ξ isin L2(Ft Rd) it holds Xsηr = X
tξr r isin [s T ] for
η = Xtξs This flow property should give rise to a PDE with a solution V (t ξ) =
E[(XtξT P
XtξT
)] but the fact that ξ has to belong to L2(Ft Rd) has the conse-
quence that V (t ξ) has to be considered over the Hilbert space L2(FRd) which
826 BUCKDAHN LI PENG AND RAINER
because of the absence of second-order Freacutechet differentiability even in simplestexamples makes such PDE difficult to handle As an alternative we associate withthe above SDE for Xtξ the equation
dXtxξs = σ
(Xtxξ
s PX
tξs
)dBs + b
(Xtxξ
s PX
tξs
)ds
(14)s isin [t T ]Xtxξ
t = x isin Rd
It turns out (see Lemma 31) that XtxPξs = X
txξs s isin [t T ] depends on ξ isin
L2(Ft Rd) only through its law Pξ Xtξs = X
txPξs |x=ξ and (X
txPξs X
tξs )0 le
t le s le T ξ isin L2(Ft Rd) has the flow propertyThe objective of our manuscript is to study under appropriate regularity as-
sumptions on the coefficients the second-order PDE which is associated with thisstochastic flow that is the PDE whose unique classical solution is given by thefunction
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(15)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
The function V is defined over [0 T ] times Rd times P2(R
d) and so the study of thefirst- and the second-order derivatives with respect to the probability law will playa crucial role In our work we have based ourselves on the notion of derivative ofa function f P2(R
d) rarr R with respect to a probability measure μ which wasstudied by Lions in his course at Collegravege de France [17] (the reader is also referredto the notes on this course redacted by Cardaliaguet [6]) The derivative of f withrespect to μ is a function partμf P2(R
d) times Rd rarr R
d (see Section 2) The mainresult of our work says that if the coefficients b and σ are twice differentiablein (xμ) with bounded Lipschitz derivatives of first and second order then thefunction V (t xPξ ) defined above is the unique classical solution of the followingnonlocal PDE of mean-field type (see Theorem 72)
parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )(16)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
= 0
V (T xPξ ) = (xPξ )
MEAN-FIELD SDES AND ASSOCIATED PDES 827
with (t xPξ ) isin [0 T ]timesRd timesP2(R
d) We see in particular that in contrast to theclassical case the derivative partμV (t xPξ y) and as a second-order derivative thederivative of partμV (t xPξ y) with respect to y are involved
Mean-field SDEs also known as McKeanndashVlasov equations were discussedthe first time by Kac [13 14] in the frame of his study of the Boltzman equa-tion for the particle density in diluted monatomic gases as well as in that of thestochastic toy model for the Vlasov kinetic equation of plasma A by now classi-cal method of solving mean-field SDEs by approximation consists in the use ofso-called N -particle systems with weak interaction formed by N equations drivenby independent Brownian motions The convergence of this system to the mean-field SDE is called in the literature propagation of chaos for the McKeanndashVlasovequation
The pioneering works by Kac and in the aftermath by other authors have at-tracted a lot of researchers interested in the study of the chaos propagation andthe limit equations in different frameworks for getting an impression we refer thereader for instance to [1 3 10ndash12 15 18ndash20] and [21] as well as the referencestherein With the pioneering works on mean-field stochastic differential games byLasry and Lions (we refer to [16] and the papers cited therein but also to [6]) newimpulses and new applications for mean-field problems were given So recentlyBuckdahn Djehiche Li and Peng [4] studied a special mean field problem by apurely stochastic approach and deduced a new kind of backward SDE (BSDE)which they called mean-field BSDE they showed that the BSDE can be obtainedby an approximation involving N -particle systems with weak interaction Theycompleted these studies of the approximation with associating a kind of centrallimit theorem for the approximating systems and obtained as limit some forward-backward SDE of mean-field type which is not only governed by a Brownianmotion but also by an independent Gaussian field In [5] deepening the investi-gation of mean-field SDEs and associated mean-field BSDEs Buckdahn Li andPeng generalized their previous results on mean-field BSDEs and in a ldquoMarko-vianrdquo framework in which the initial data (t x) were frozen in the law variable ofthe coefficients they investigated the associated nonlocal PDE However our ob-jective has been to overcome this partial freezing of initial data in the mean-fieldSDE and to study the associated PDE and this is done in our present manuscriptOur approach is highly inspired by the courses given by Lions [17] at Collegravege deFrance (redacted by Cardaliaguet [6]) and by recent works of Bensoussan FrehseYam [2] and Carmona and Delarue who directly inspired by the works of LasryLions [16] and the courses of Lions [17] translated his rather analytical approachinto a stochastic one let us cite [8 9] and the references indicated therein
Let us describe the organization of our manuscript and link this description withthe explanation of the novelty in our approach
In Section 2 ldquoPreliminariesrdquo we introduce the framework of our study Partic-ular attention is paid to a recall of Lionsrsquo definition of the derivative of a function
828 BUCKDAHN LI PENG AND RAINER
defined over the space of probability measures over Rd with finite second mo-
ment On the basis of this notion of first-order derivatives we introduce in exactlythe same spirit the second-order derivatives of such a function The first- and thesecond-order differentiability of a function with respect to a probability measureallows in the following to derive a second-order Taylor expansion (Lemma 21)which is new and turns out to be crucial in our approach We give an example(Example 23) which shows that in general even in the most regular cases thefunctions lifted from the space of probability measures with finite second momentto the space of square integrable random variables are not twice Freacutechet differen-tiable on L2 but well twice differentiable with respect to the probability measureTo the best of our knowledge the passage to higher order derivatives with respectto the measure the second-order Taylor expansion with respect to the measure andExample 23 are new They constitute the crucial paves in our approach in par-ticular also for the derivation of the Itocirc formula for mean-field Itocirc processes (seeTheorem 71 in Section 7)
In Section 3 we introduce our mean-field SDE with our standard assumptionson its coefficients [their twice fold differentiability with respect to (xμ) withbounded Lipschitz derivatives of first and second order] and we study useful prop-erties of the mean-field SDE A crucial step in our approach constitutes the splittingof mean-field SDE (11) into (13) and (14)mdasha system of mean-field SDEs whichunlike (11) generates a flow The flow concerns the couple formed by the stateand by the law of the process Such a splitting for the study of mean-field SDEs isnovel
A central property studied in Section 4 is the differentiability of its solutionprocess XtxPξ with respect to the probability law Pξ The identification of thederivative of XtxPξ with respect to the probability law and the equation satisfiedby it are new to the best of our knowledge The investigations of Section 4 are com-pleted by Section 5 which is devoted to the study of the second-order derivativesof XtxPξ and so namely for that with respect to the probability law The first- andthe second-order derivatives of XtxPξ are characterized as the unique solution ofassociated SDEs which on their part allow to get estimates for the derivatives oforder 1 and 2 of XtxPξ The results obtained for the process XtxPξ and so alsofor Xtξ in the Sections 3ndash5 are used in Section 6 for the proof of the regularity ofthe value function V (t xPξ ) Finally Section 7 is devoted to a novel Itocirc formulaassociated with mean-field problems and it gives our main result Theorem 72stating that our value function V is the unique classical solution of the PDE (16)of mean-field type given above
2 Preliminaries Let us begin with introducing some notation and con-cepts which we will need in our further computations We shall in particularintroduce the notion of differentiability of a function f defined over the spaceP2(R
d) of all probability measures μ over (RdB(Rd)) with finite second mo-ment
intRd |x|2μ(dx) lt infin [B(Rd) denotes the Borel σ -field over Rd ] The space
MEAN-FIELD SDES AND ASSOCIATED PDES 829
P2(R2d) is endowed with the 2-Wasserstein metric
W2(μ ν) = inf(int
RdtimesRd|x minus y|2ρ(dx dy)
)12
ρ isin P2(R
2d)(21)
with ρ(middot timesR
d) = μρ(R
d times middot) = ν
μ ν isin P2
(R
d)
Among the different notions of differentiability of a function f defined overP2(R
d) we adopt for our approach that introduced by Lions [17] in his lectures atCollegravege de France in Paris and revised in the notes by Cardaliaguet [6] we referthe reader also for instance to Carmona and Delarue [9] Let us consider a proba-bility space (FP ) which is ldquorich enoughrdquo (the precise space we will work withwill be introduced later) ldquoRich enoughrdquo means that for every μ isin P2(R
d) thereis a random variable ϑ isin L2(FRd)(= L2(FP Rd)) such that Pϑ = μ It iswell known that the probability space ([01]B([01]) dx) has this property
Identifying the random variables in L2(FRd) which coincide P -as we canregard L2(FRd) as a Hilbert space with inner product (ξ η)L2 = E[ξ middot η] ξ η isinL2(FRd) and norm |ξ |L2 = (ξ ξ)
12L2 Recall that due to the definition made
by Lions [6] (see Cardaliaguet [7]) a function f P2(Rd) rarr R is said to be dif-
ferentiable in μ isin P2(Rd) if for f (ϑ) = f (Pϑ)ϑ isin L2(FRd) there is some
ϑ0 isin L2(FRd) with Pϑ0 = μ such that the function f L2(FRd) rarr R is dif-ferentiable (in Freacutechet sense) in ϑ0 that is there exists a linear continuous mappingDf (ϑ0) L2(FRd) rarrR (Df (ϑ0) isin L(L2(FRd)R)) such that
f (ϑ0 + η) minus f (ϑ0) = Df (ϑ0)(η) + o(|η|L2
)(22)
with |η|L2 rarr 0 for η isin L2(FRd) Since Df (ϑ0) isin L(L2(FRd)R) Rieszrsquo rep-resentation theorem yields the existence of a (P -as) unique random variable θ0 isinL2(FRd) such that Df (ϑ0)(η) = (θ0 η)L2 = E[θ0η] for all η isin L2(FRd) In[17] (see also [6]) it has been proved that there is a Borel function h0 Rd rarr R
d
such that θ0 = h0(ϑ0)P -as The function h0 only depends on the law Pϑ0 but noton ϑ0 itself Taking into account the definition of f this allows to write
f (Pϑ) minus f (Pϑ0) = E[h0(ϑ0) middot (ϑ minus ϑ0)
] + o(|ϑ minus ϑ0|L2
)(23)
ϑ isin L2(FRd)We call partμf (Pϑ0 y) = h0(y) y isin R
d the derivative of f P2(Rd) rarr R at
Pϑ0 Note that partμf (Pϑ0 y) is only Pϑ0(dy)-ae uniquely determinedHowever in our approach we have to consider functions f P2(R
d) rarrR whichare differentiable in all elements of P2(R
d) In order to simplify the argumentwe suppose that f L2(FRd) rarr R is Freacutechet differential over the whole spaceL2(FRd) This corresponds to a large class of important examples In this casewe have the derivative partμf (Pϑ y) defined Pϑ(dy)-ae for all ϑ isin L2(FRd)In Lemma 32 [9] it is shown that if furthermore the Freacutechet derivative Df L2(FRd) rarr L(L2(FRd)R) is Lipschitz continuous (with a Lipschitz constant
830 BUCKDAHN LI PENG AND RAINER
K isin R+) then there is for all ϑ isin L2(FRd) a Pϑ -version of partμf (Pϑ middot) Rd rarrR
d such that∣∣partμf (Pϑ y) minus partμf(Pϑ yprime)∣∣ le K
∣∣y minus yprime∣∣ for all y yprime isin Rd
This motivates us to make the following definition
DEFINITION 21 We say that f isin C11b (P2(R
d)) [continuously differentiableover P2(R
d) with Lipschitz-continuous bounded derivative] if there exists for allϑ isin L2(FRd) a Pϑ -modification of partμf (Pϑ middot) again denoted by partμf (Pϑ middot)such that partμf P2(R
d) timesRd rarr R
d is bounded and Lipschitz continuous that isthere is some real constant C such that
(i) |partμf (μx)| le Cμ isin P2(Rd) x isin R
d (ii) |partμf (μx) minus partμf (μprime xprime)| le C(W2(μμprime) + |x minus xprime|)μμprime isin P2(R
d) xxprime isin R
d
we consider this function partμf as the derivative of f
REMARK 21 Let us point out that if f isin C11b (P2(R
d)) the version ofpartμf (Pϑ middot) ϑ isin L2(FRd) indicated in Definition 21 is unique Indeed givenϑ isin L2(FRd) let θ be a d-dimensional vector of independent standard nor-mally distributed random variables which are independent of ϑ Then sincepartμf (Pϑ+εθ ϑ + εθ) is only P -as defined partμf (Pϑ+εθ y) is determined onlyPϑ+εθ (dy)-as Observing that the random variable ϑ +εθ possesses a strictly pos-itive density on R
d it follows that Pϑ+εθ and the Lebesgue measure over Rd areequivalent Consequently partμf (Pϑ+εθ y) is defined dy-ae on R
d From the Lip-schitz continuity (ii) of partμf in Definition 21 it then follows that partμf (Pϑ+εθ y)
is defined for all y isin Rd and taking the limit 0 lt ε darr 0 yields that partμf (Pϑ y) is
uniquely defined for all y isin Rd
Given now f isin C11b (P2(R
d)) for fixed y isin Rd the question of the differen-
tiability of its components (partμf )j (middot y) P2(Rd) rarr R1 le j le d raises and
it can be discussed in the same way as the first-order derivative partμf above If(partμf )j (middot y) P2(R
d) rarr R belongs to C11b (P2(R
d)) we have that its derivativepartμ((partμf )j (middot y))(middot middot) P2(R
d)timesRd rarrR
d is a Lipschitz-continuous function forevery y isin R
d Then
part2μf (μx y) = (
partμ
((partμf )j (middot y)
)(μ x)
)1lejled
(24)(μ x y) isin P2
(R
d) timesR
d timesRd
defines a function part2μf P2(R
d) timesRd timesR
d rarrRd otimesR
d
DEFINITION 22 We say that f isin C21b (P2(R
d)) if f isin C11b (P2(R
d)) and
MEAN-FIELD SDES AND ASSOCIATED PDES 831
(i) (partμf )j (middot y) isin C11b (P2(R
d)) for all y isin Rd1 le j le d and part2
μf P2(R
d) timesRd timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous(ii) partμf (μ middot) Rd rarr R
d is differentiable for every μ isin P2(Rd) and its
derivative partypartμf P2(Rd)timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous
Adopting the above introduced notation we consider a functionf isin C
21b (P2(R
d)) and discuss its second-order Taylor expansion For this endwe have still to introduce some notation Let ( F P ) be a copy of the prob-ability space (FP ) For any random variable (of arbitrary dimension) ϑ
over (FP ) we denote by ϑ a copy (of the same law as ϑ but definedover ( F P ) Pϑ = Pϑ The expectation E[middot] = int
(middot) dP acts only over thevariables endowed with a tilde This can be made rigorous by working withthe product space (FP ) otimes ( F P ) = (FP ) otimes (FP ) and puttingϑ(ωω) = ϑ(ω) (ωω) isin times = times for ϑ random variable defined over(FP )) Of course this formalism can be easily extended from random vari-ables to stochastic processes
With the above notation and writing a otimes b = (aibj )1leijled for a = (ai)1leiled
b = (bj )1lejled isin Rd we can state now the following result
LEMMA 21 Let f isin C21b (P2(R
d)) Then for any given ϑ0 isin L2(FRd) wehave the following second-order expansion
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](25)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R(PϑPϑ0)
ϑ isin L2(FRd
)
where η = ϑ minus ϑ0 and for all ϑ isin L2(FRd) the remainder R(PϑPϑ0) satisfiesthe estimate ∣∣R(PϑPϑ0)
∣∣ le C((
E[|ϑ minus ϑ0|2])32 + E
[|ϑ minus ϑ0|3])(26)
le CE[|ϑ minus ϑ0|3]
The constant C isin R+ only depends on the Lipschitz constants of part2μf and partypartμf
We observe that the above second-order expansion does not constitute a second-order Taylor expansion for the associated function f L2(FRd) rarr R since theremainder term E[|ϑ minus ϑ0|3 and |ϑ minus ϑ0|2] is not of order o(|ϑ minus ϑ0|2L2) Indeed asthe following example shows in general we only have E[|ϑ minusϑ0|3 and |ϑ minusϑ0|2] =O(|ϑ minus ϑ0|2L2)
832 BUCKDAHN LI PENG AND RAINER
EXAMPLE 21 Let ϑ = IA with A isin F such that P(A) rarr 0 as rarr infin
Then ϑ rarr 0 in L2( rarr infin) and E[|ϑ|3 and |ϑ|2] = P(A) = E[ϑ2 ] = |ϑ|2L2 rarr
0 ( rarr infin)
If we had in Lemma 21 a remainder R(PϑPϑ0) = o(|ϑ minus ϑ0|2L2) this wouldsuggest that f (ζ ) = f (Pζ ) is twice Freacutechet differentiable at ϑ0 But however asthe following Example 23 shows even in easiest cases f is not twice Freacutechetdifferentiable
However for our purposes the above expansion is fine
PROOF OF LEMMA 21 Let ϑ0 isin L2(FRd) Then for all ϑ isin L2(FRd)putting η = ϑ minus ϑ0 and using the fact that f isin C
21b (P2(R
d)) we have
f (Pϑ) minus f (Pϑ0) =int 1
0
d
dλf (Pϑ0+λη) dλ
(27)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
Let us now compute ddλ
partμf (Pϑ0+ληϑ0 +λη) Since f isin C21b (P2(R
d)) the liftedfunction partμf (ξ y) = partμf (Pξ y) ξ isin L2(FRd) is Freacutechet differentiable in ξ and
d
dλpartμf (Pϑ0+λη y) = d
dλpartμf (ϑ0 + λη y)
(28)= E
[part2μf (Pϑ0+ληϑ0 + λη y) middot η]
λ isin R y isin Rd Then choosing an independent copy (ϑ ϑ0) of (ϑϑ0) defined
over ( F P ) (ie in particular P(ϑϑ0)= P(ϑϑ0)) we have for η = ϑ minus ϑ0
d
dλpartμf (Pϑ0+λη y) = E
[part2μf (Pϑ0+λη ϑ0 + λη y) middot η]
(29)λ isin R y isin R
d
Consequently
d
dλpartμf (Pϑ0+ληϑ0 + λη) = E
[part2μf (Pϑ0+λη ϑ0 + ληϑ0 + λη) middot η]
(210)+ partypartμf (Pϑ0+ληϑ0 + λη) middot η
Thus
f (Pϑ) minus f (Pϑ0)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
MEAN-FIELD SDES AND ASSOCIATED PDES 833
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[d
dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η
]dρ dλ(211)
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[tr
(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η
)]dρ dλ
+int 1
0
int λ
0E
[E
[tr
(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η
)]]dρ dλ
From this latter relation we get
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](212)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R1(PϑPϑ0) + R2(PϑPϑ0)
with the remainders
R1(PϑPϑ0) =int 1
0
int λ
0E
[E
[tr
((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)
(213)minus part2
μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η
)]]dρ dλ
and
R2(PϑPϑ0) =int 1
0
int λ
0E
[tr
((partypartμf (Pϑ0+ρηϑ0 + ρη)
(214)minus partypartμf (Pϑ0 ϑ0)
) middot η otimes η)]
dρ dλ
Finally from the boundedness and the Lipschitz continuity of the functions part2μf
and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf
|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement
Let us finish our preliminary discussion with two illustrating examples
EXAMPLE 22 Given two twice continuously differentiable functions h R
d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and
f (ϑ0 + η) minus f (ϑ0) =int 1
0gprime(E[
h(ϑ0 + sη)])
E[hprime(ϑ0 + sη)η
]ds
= gprime(E[h(ϑ0)
])E
[hprime(ϑ0)η
] + o(|η|L2
)= E
[gprime(E[
h(ϑ0)])
hprime(ϑ0)η] + o
(|η|L2)
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
MEAN-FIELD SDES AND ASSOCIATED PDES 825
7 Itocirc formula and PDE associated with mean-field SDE 870Acknowledgments 877References 877
1 Introduction Given a complete probability space (FP ) endowedwith a Brownian motion B = (Bt )tisin[0T ] and its filtration F = (Ft )tisin[0T ] aug-mented by all P -null sets and a sufficiently rich sub-σ -algebra F0 independentof B we consider the mean-field stochastic differential equation (SDE) alsoknown under the name of McKeanndashVlasov SDE
dXtxs = σ
(Xtx
s PXtxs
)dBs + b
(Xtx
s PXtxs
)ds
(11)s isin [t T ]Xtx
t = x isinRd
It is well known that under an appropriate Lipschitz assumption on the coeffi-cients this equation possesses for all (t x) isin [t T ]timesR
d a unique solution Xtxs s isin
[0 T ] For the classical SDE whose coefficients σ(xμ) = σ(x) b(xμ) = b(x)
depend only on x isin Rd but not on the probability measure μ it is well known
that the solution Xtxs 0 le t le s le T x isinR
d defines a flow and if the coefficientsare regular enough the unique classical solution of the partial differential equation(PDE)
parttV (t x) + 12 tr
(σσ lowast(x)D2
xV (t x)) + b(x)DxV (t x) = 0
(t x) isin [0 T ] timesRd(12)
V (T x) = (x) x isin Rd
is V (t x) = E[(XtxT )] (t x) isin [0 T ] times R
d But how about the above SDEwhose coefficients depend on (xμ) isin R
d times P2(Rd) where P2(R
d) denotes thespace of the probability measures over Rd with finite second moment Of coursefor an SDE with coefficients depending on (xμ) the solution Xtx
s 0 le s le t leT x isin R
d does obviously not define a flow But we see easily that if we replacethe deterministic initial condition X
txt = x isin R
d by a square integrable randomvariable X
tξt = ξ isin L2(Ft Rd)(= L2(Ft P Rd)) and consider the SDE
dXtξs = σ
(Xtξ
s PX
tξs
)dBs + b
(Xtξ
s PX
tξs
)ds
(13)s isin [t T ]Xtξ
t = ξ isin Rd
(where obviously in general Xtξ = Xtx |x=ξ ) then we have the flow property
For all 0 le t le s le T and ξ isin L2(Ft Rd) it holds Xsηr = X
tξr r isin [s T ] for
η = Xtξs This flow property should give rise to a PDE with a solution V (t ξ) =
E[(XtξT P
XtξT
)] but the fact that ξ has to belong to L2(Ft Rd) has the conse-
quence that V (t ξ) has to be considered over the Hilbert space L2(FRd) which
826 BUCKDAHN LI PENG AND RAINER
because of the absence of second-order Freacutechet differentiability even in simplestexamples makes such PDE difficult to handle As an alternative we associate withthe above SDE for Xtξ the equation
dXtxξs = σ
(Xtxξ
s PX
tξs
)dBs + b
(Xtxξ
s PX
tξs
)ds
(14)s isin [t T ]Xtxξ
t = x isin Rd
It turns out (see Lemma 31) that XtxPξs = X
txξs s isin [t T ] depends on ξ isin
L2(Ft Rd) only through its law Pξ Xtξs = X
txPξs |x=ξ and (X
txPξs X
tξs )0 le
t le s le T ξ isin L2(Ft Rd) has the flow propertyThe objective of our manuscript is to study under appropriate regularity as-
sumptions on the coefficients the second-order PDE which is associated with thisstochastic flow that is the PDE whose unique classical solution is given by thefunction
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(15)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
The function V is defined over [0 T ] times Rd times P2(R
d) and so the study of thefirst- and the second-order derivatives with respect to the probability law will playa crucial role In our work we have based ourselves on the notion of derivative ofa function f P2(R
d) rarr R with respect to a probability measure μ which wasstudied by Lions in his course at Collegravege de France [17] (the reader is also referredto the notes on this course redacted by Cardaliaguet [6]) The derivative of f withrespect to μ is a function partμf P2(R
d) times Rd rarr R
d (see Section 2) The mainresult of our work says that if the coefficients b and σ are twice differentiablein (xμ) with bounded Lipschitz derivatives of first and second order then thefunction V (t xPξ ) defined above is the unique classical solution of the followingnonlocal PDE of mean-field type (see Theorem 72)
parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )(16)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
= 0
V (T xPξ ) = (xPξ )
MEAN-FIELD SDES AND ASSOCIATED PDES 827
with (t xPξ ) isin [0 T ]timesRd timesP2(R
d) We see in particular that in contrast to theclassical case the derivative partμV (t xPξ y) and as a second-order derivative thederivative of partμV (t xPξ y) with respect to y are involved
Mean-field SDEs also known as McKeanndashVlasov equations were discussedthe first time by Kac [13 14] in the frame of his study of the Boltzman equa-tion for the particle density in diluted monatomic gases as well as in that of thestochastic toy model for the Vlasov kinetic equation of plasma A by now classi-cal method of solving mean-field SDEs by approximation consists in the use ofso-called N -particle systems with weak interaction formed by N equations drivenby independent Brownian motions The convergence of this system to the mean-field SDE is called in the literature propagation of chaos for the McKeanndashVlasovequation
The pioneering works by Kac and in the aftermath by other authors have at-tracted a lot of researchers interested in the study of the chaos propagation andthe limit equations in different frameworks for getting an impression we refer thereader for instance to [1 3 10ndash12 15 18ndash20] and [21] as well as the referencestherein With the pioneering works on mean-field stochastic differential games byLasry and Lions (we refer to [16] and the papers cited therein but also to [6]) newimpulses and new applications for mean-field problems were given So recentlyBuckdahn Djehiche Li and Peng [4] studied a special mean field problem by apurely stochastic approach and deduced a new kind of backward SDE (BSDE)which they called mean-field BSDE they showed that the BSDE can be obtainedby an approximation involving N -particle systems with weak interaction Theycompleted these studies of the approximation with associating a kind of centrallimit theorem for the approximating systems and obtained as limit some forward-backward SDE of mean-field type which is not only governed by a Brownianmotion but also by an independent Gaussian field In [5] deepening the investi-gation of mean-field SDEs and associated mean-field BSDEs Buckdahn Li andPeng generalized their previous results on mean-field BSDEs and in a ldquoMarko-vianrdquo framework in which the initial data (t x) were frozen in the law variable ofthe coefficients they investigated the associated nonlocal PDE However our ob-jective has been to overcome this partial freezing of initial data in the mean-fieldSDE and to study the associated PDE and this is done in our present manuscriptOur approach is highly inspired by the courses given by Lions [17] at Collegravege deFrance (redacted by Cardaliaguet [6]) and by recent works of Bensoussan FrehseYam [2] and Carmona and Delarue who directly inspired by the works of LasryLions [16] and the courses of Lions [17] translated his rather analytical approachinto a stochastic one let us cite [8 9] and the references indicated therein
Let us describe the organization of our manuscript and link this description withthe explanation of the novelty in our approach
In Section 2 ldquoPreliminariesrdquo we introduce the framework of our study Partic-ular attention is paid to a recall of Lionsrsquo definition of the derivative of a function
828 BUCKDAHN LI PENG AND RAINER
defined over the space of probability measures over Rd with finite second mo-
ment On the basis of this notion of first-order derivatives we introduce in exactlythe same spirit the second-order derivatives of such a function The first- and thesecond-order differentiability of a function with respect to a probability measureallows in the following to derive a second-order Taylor expansion (Lemma 21)which is new and turns out to be crucial in our approach We give an example(Example 23) which shows that in general even in the most regular cases thefunctions lifted from the space of probability measures with finite second momentto the space of square integrable random variables are not twice Freacutechet differen-tiable on L2 but well twice differentiable with respect to the probability measureTo the best of our knowledge the passage to higher order derivatives with respectto the measure the second-order Taylor expansion with respect to the measure andExample 23 are new They constitute the crucial paves in our approach in par-ticular also for the derivation of the Itocirc formula for mean-field Itocirc processes (seeTheorem 71 in Section 7)
In Section 3 we introduce our mean-field SDE with our standard assumptionson its coefficients [their twice fold differentiability with respect to (xμ) withbounded Lipschitz derivatives of first and second order] and we study useful prop-erties of the mean-field SDE A crucial step in our approach constitutes the splittingof mean-field SDE (11) into (13) and (14)mdasha system of mean-field SDEs whichunlike (11) generates a flow The flow concerns the couple formed by the stateand by the law of the process Such a splitting for the study of mean-field SDEs isnovel
A central property studied in Section 4 is the differentiability of its solutionprocess XtxPξ with respect to the probability law Pξ The identification of thederivative of XtxPξ with respect to the probability law and the equation satisfiedby it are new to the best of our knowledge The investigations of Section 4 are com-pleted by Section 5 which is devoted to the study of the second-order derivativesof XtxPξ and so namely for that with respect to the probability law The first- andthe second-order derivatives of XtxPξ are characterized as the unique solution ofassociated SDEs which on their part allow to get estimates for the derivatives oforder 1 and 2 of XtxPξ The results obtained for the process XtxPξ and so alsofor Xtξ in the Sections 3ndash5 are used in Section 6 for the proof of the regularity ofthe value function V (t xPξ ) Finally Section 7 is devoted to a novel Itocirc formulaassociated with mean-field problems and it gives our main result Theorem 72stating that our value function V is the unique classical solution of the PDE (16)of mean-field type given above
2 Preliminaries Let us begin with introducing some notation and con-cepts which we will need in our further computations We shall in particularintroduce the notion of differentiability of a function f defined over the spaceP2(R
d) of all probability measures μ over (RdB(Rd)) with finite second mo-ment
intRd |x|2μ(dx) lt infin [B(Rd) denotes the Borel σ -field over Rd ] The space
MEAN-FIELD SDES AND ASSOCIATED PDES 829
P2(R2d) is endowed with the 2-Wasserstein metric
W2(μ ν) = inf(int
RdtimesRd|x minus y|2ρ(dx dy)
)12
ρ isin P2(R
2d)(21)
with ρ(middot timesR
d) = μρ(R
d times middot) = ν
μ ν isin P2
(R
d)
Among the different notions of differentiability of a function f defined overP2(R
d) we adopt for our approach that introduced by Lions [17] in his lectures atCollegravege de France in Paris and revised in the notes by Cardaliaguet [6] we referthe reader also for instance to Carmona and Delarue [9] Let us consider a proba-bility space (FP ) which is ldquorich enoughrdquo (the precise space we will work withwill be introduced later) ldquoRich enoughrdquo means that for every μ isin P2(R
d) thereis a random variable ϑ isin L2(FRd)(= L2(FP Rd)) such that Pϑ = μ It iswell known that the probability space ([01]B([01]) dx) has this property
Identifying the random variables in L2(FRd) which coincide P -as we canregard L2(FRd) as a Hilbert space with inner product (ξ η)L2 = E[ξ middot η] ξ η isinL2(FRd) and norm |ξ |L2 = (ξ ξ)
12L2 Recall that due to the definition made
by Lions [6] (see Cardaliaguet [7]) a function f P2(Rd) rarr R is said to be dif-
ferentiable in μ isin P2(Rd) if for f (ϑ) = f (Pϑ)ϑ isin L2(FRd) there is some
ϑ0 isin L2(FRd) with Pϑ0 = μ such that the function f L2(FRd) rarr R is dif-ferentiable (in Freacutechet sense) in ϑ0 that is there exists a linear continuous mappingDf (ϑ0) L2(FRd) rarrR (Df (ϑ0) isin L(L2(FRd)R)) such that
f (ϑ0 + η) minus f (ϑ0) = Df (ϑ0)(η) + o(|η|L2
)(22)
with |η|L2 rarr 0 for η isin L2(FRd) Since Df (ϑ0) isin L(L2(FRd)R) Rieszrsquo rep-resentation theorem yields the existence of a (P -as) unique random variable θ0 isinL2(FRd) such that Df (ϑ0)(η) = (θ0 η)L2 = E[θ0η] for all η isin L2(FRd) In[17] (see also [6]) it has been proved that there is a Borel function h0 Rd rarr R
d
such that θ0 = h0(ϑ0)P -as The function h0 only depends on the law Pϑ0 but noton ϑ0 itself Taking into account the definition of f this allows to write
f (Pϑ) minus f (Pϑ0) = E[h0(ϑ0) middot (ϑ minus ϑ0)
] + o(|ϑ minus ϑ0|L2
)(23)
ϑ isin L2(FRd)We call partμf (Pϑ0 y) = h0(y) y isin R
d the derivative of f P2(Rd) rarr R at
Pϑ0 Note that partμf (Pϑ0 y) is only Pϑ0(dy)-ae uniquely determinedHowever in our approach we have to consider functions f P2(R
d) rarrR whichare differentiable in all elements of P2(R
d) In order to simplify the argumentwe suppose that f L2(FRd) rarr R is Freacutechet differential over the whole spaceL2(FRd) This corresponds to a large class of important examples In this casewe have the derivative partμf (Pϑ y) defined Pϑ(dy)-ae for all ϑ isin L2(FRd)In Lemma 32 [9] it is shown that if furthermore the Freacutechet derivative Df L2(FRd) rarr L(L2(FRd)R) is Lipschitz continuous (with a Lipschitz constant
830 BUCKDAHN LI PENG AND RAINER
K isin R+) then there is for all ϑ isin L2(FRd) a Pϑ -version of partμf (Pϑ middot) Rd rarrR
d such that∣∣partμf (Pϑ y) minus partμf(Pϑ yprime)∣∣ le K
∣∣y minus yprime∣∣ for all y yprime isin Rd
This motivates us to make the following definition
DEFINITION 21 We say that f isin C11b (P2(R
d)) [continuously differentiableover P2(R
d) with Lipschitz-continuous bounded derivative] if there exists for allϑ isin L2(FRd) a Pϑ -modification of partμf (Pϑ middot) again denoted by partμf (Pϑ middot)such that partμf P2(R
d) timesRd rarr R
d is bounded and Lipschitz continuous that isthere is some real constant C such that
(i) |partμf (μx)| le Cμ isin P2(Rd) x isin R
d (ii) |partμf (μx) minus partμf (μprime xprime)| le C(W2(μμprime) + |x minus xprime|)μμprime isin P2(R
d) xxprime isin R
d
we consider this function partμf as the derivative of f
REMARK 21 Let us point out that if f isin C11b (P2(R
d)) the version ofpartμf (Pϑ middot) ϑ isin L2(FRd) indicated in Definition 21 is unique Indeed givenϑ isin L2(FRd) let θ be a d-dimensional vector of independent standard nor-mally distributed random variables which are independent of ϑ Then sincepartμf (Pϑ+εθ ϑ + εθ) is only P -as defined partμf (Pϑ+εθ y) is determined onlyPϑ+εθ (dy)-as Observing that the random variable ϑ +εθ possesses a strictly pos-itive density on R
d it follows that Pϑ+εθ and the Lebesgue measure over Rd areequivalent Consequently partμf (Pϑ+εθ y) is defined dy-ae on R
d From the Lip-schitz continuity (ii) of partμf in Definition 21 it then follows that partμf (Pϑ+εθ y)
is defined for all y isin Rd and taking the limit 0 lt ε darr 0 yields that partμf (Pϑ y) is
uniquely defined for all y isin Rd
Given now f isin C11b (P2(R
d)) for fixed y isin Rd the question of the differen-
tiability of its components (partμf )j (middot y) P2(Rd) rarr R1 le j le d raises and
it can be discussed in the same way as the first-order derivative partμf above If(partμf )j (middot y) P2(R
d) rarr R belongs to C11b (P2(R
d)) we have that its derivativepartμ((partμf )j (middot y))(middot middot) P2(R
d)timesRd rarrR
d is a Lipschitz-continuous function forevery y isin R
d Then
part2μf (μx y) = (
partμ
((partμf )j (middot y)
)(μ x)
)1lejled
(24)(μ x y) isin P2
(R
d) timesR
d timesRd
defines a function part2μf P2(R
d) timesRd timesR
d rarrRd otimesR
d
DEFINITION 22 We say that f isin C21b (P2(R
d)) if f isin C11b (P2(R
d)) and
MEAN-FIELD SDES AND ASSOCIATED PDES 831
(i) (partμf )j (middot y) isin C11b (P2(R
d)) for all y isin Rd1 le j le d and part2
μf P2(R
d) timesRd timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous(ii) partμf (μ middot) Rd rarr R
d is differentiable for every μ isin P2(Rd) and its
derivative partypartμf P2(Rd)timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous
Adopting the above introduced notation we consider a functionf isin C
21b (P2(R
d)) and discuss its second-order Taylor expansion For this endwe have still to introduce some notation Let ( F P ) be a copy of the prob-ability space (FP ) For any random variable (of arbitrary dimension) ϑ
over (FP ) we denote by ϑ a copy (of the same law as ϑ but definedover ( F P ) Pϑ = Pϑ The expectation E[middot] = int
(middot) dP acts only over thevariables endowed with a tilde This can be made rigorous by working withthe product space (FP ) otimes ( F P ) = (FP ) otimes (FP ) and puttingϑ(ωω) = ϑ(ω) (ωω) isin times = times for ϑ random variable defined over(FP )) Of course this formalism can be easily extended from random vari-ables to stochastic processes
With the above notation and writing a otimes b = (aibj )1leijled for a = (ai)1leiled
b = (bj )1lejled isin Rd we can state now the following result
LEMMA 21 Let f isin C21b (P2(R
d)) Then for any given ϑ0 isin L2(FRd) wehave the following second-order expansion
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](25)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R(PϑPϑ0)
ϑ isin L2(FRd
)
where η = ϑ minus ϑ0 and for all ϑ isin L2(FRd) the remainder R(PϑPϑ0) satisfiesthe estimate ∣∣R(PϑPϑ0)
∣∣ le C((
E[|ϑ minus ϑ0|2])32 + E
[|ϑ minus ϑ0|3])(26)
le CE[|ϑ minus ϑ0|3]
The constant C isin R+ only depends on the Lipschitz constants of part2μf and partypartμf
We observe that the above second-order expansion does not constitute a second-order Taylor expansion for the associated function f L2(FRd) rarr R since theremainder term E[|ϑ minus ϑ0|3 and |ϑ minus ϑ0|2] is not of order o(|ϑ minus ϑ0|2L2) Indeed asthe following example shows in general we only have E[|ϑ minusϑ0|3 and |ϑ minusϑ0|2] =O(|ϑ minus ϑ0|2L2)
832 BUCKDAHN LI PENG AND RAINER
EXAMPLE 21 Let ϑ = IA with A isin F such that P(A) rarr 0 as rarr infin
Then ϑ rarr 0 in L2( rarr infin) and E[|ϑ|3 and |ϑ|2] = P(A) = E[ϑ2 ] = |ϑ|2L2 rarr
0 ( rarr infin)
If we had in Lemma 21 a remainder R(PϑPϑ0) = o(|ϑ minus ϑ0|2L2) this wouldsuggest that f (ζ ) = f (Pζ ) is twice Freacutechet differentiable at ϑ0 But however asthe following Example 23 shows even in easiest cases f is not twice Freacutechetdifferentiable
However for our purposes the above expansion is fine
PROOF OF LEMMA 21 Let ϑ0 isin L2(FRd) Then for all ϑ isin L2(FRd)putting η = ϑ minus ϑ0 and using the fact that f isin C
21b (P2(R
d)) we have
f (Pϑ) minus f (Pϑ0) =int 1
0
d
dλf (Pϑ0+λη) dλ
(27)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
Let us now compute ddλ
partμf (Pϑ0+ληϑ0 +λη) Since f isin C21b (P2(R
d)) the liftedfunction partμf (ξ y) = partμf (Pξ y) ξ isin L2(FRd) is Freacutechet differentiable in ξ and
d
dλpartμf (Pϑ0+λη y) = d
dλpartμf (ϑ0 + λη y)
(28)= E
[part2μf (Pϑ0+ληϑ0 + λη y) middot η]
λ isin R y isin Rd Then choosing an independent copy (ϑ ϑ0) of (ϑϑ0) defined
over ( F P ) (ie in particular P(ϑϑ0)= P(ϑϑ0)) we have for η = ϑ minus ϑ0
d
dλpartμf (Pϑ0+λη y) = E
[part2μf (Pϑ0+λη ϑ0 + λη y) middot η]
(29)λ isin R y isin R
d
Consequently
d
dλpartμf (Pϑ0+ληϑ0 + λη) = E
[part2μf (Pϑ0+λη ϑ0 + ληϑ0 + λη) middot η]
(210)+ partypartμf (Pϑ0+ληϑ0 + λη) middot η
Thus
f (Pϑ) minus f (Pϑ0)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
MEAN-FIELD SDES AND ASSOCIATED PDES 833
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[d
dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η
]dρ dλ(211)
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[tr
(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η
)]dρ dλ
+int 1
0
int λ
0E
[E
[tr
(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η
)]]dρ dλ
From this latter relation we get
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](212)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R1(PϑPϑ0) + R2(PϑPϑ0)
with the remainders
R1(PϑPϑ0) =int 1
0
int λ
0E
[E
[tr
((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)
(213)minus part2
μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η
)]]dρ dλ
and
R2(PϑPϑ0) =int 1
0
int λ
0E
[tr
((partypartμf (Pϑ0+ρηϑ0 + ρη)
(214)minus partypartμf (Pϑ0 ϑ0)
) middot η otimes η)]
dρ dλ
Finally from the boundedness and the Lipschitz continuity of the functions part2μf
and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf
|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement
Let us finish our preliminary discussion with two illustrating examples
EXAMPLE 22 Given two twice continuously differentiable functions h R
d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and
f (ϑ0 + η) minus f (ϑ0) =int 1
0gprime(E[
h(ϑ0 + sη)])
E[hprime(ϑ0 + sη)η
]ds
= gprime(E[h(ϑ0)
])E
[hprime(ϑ0)η
] + o(|η|L2
)= E
[gprime(E[
h(ϑ0)])
hprime(ϑ0)η] + o
(|η|L2)
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
826 BUCKDAHN LI PENG AND RAINER
because of the absence of second-order Freacutechet differentiability even in simplestexamples makes such PDE difficult to handle As an alternative we associate withthe above SDE for Xtξ the equation
dXtxξs = σ
(Xtxξ
s PX
tξs
)dBs + b
(Xtxξ
s PX
tξs
)ds
(14)s isin [t T ]Xtxξ
t = x isin Rd
It turns out (see Lemma 31) that XtxPξs = X
txξs s isin [t T ] depends on ξ isin
L2(Ft Rd) only through its law Pξ Xtξs = X
txPξs |x=ξ and (X
txPξs X
tξs )0 le
t le s le T ξ isin L2(Ft Rd) has the flow propertyThe objective of our manuscript is to study under appropriate regularity as-
sumptions on the coefficients the second-order PDE which is associated with thisstochastic flow that is the PDE whose unique classical solution is given by thefunction
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(15)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
The function V is defined over [0 T ] times Rd times P2(R
d) and so the study of thefirst- and the second-order derivatives with respect to the probability law will playa crucial role In our work we have based ourselves on the notion of derivative ofa function f P2(R
d) rarr R with respect to a probability measure μ which wasstudied by Lions in his course at Collegravege de France [17] (the reader is also referredto the notes on this course redacted by Cardaliaguet [6]) The derivative of f withrespect to μ is a function partμf P2(R
d) times Rd rarr R
d (see Section 2) The mainresult of our work says that if the coefficients b and σ are twice differentiablein (xμ) with bounded Lipschitz derivatives of first and second order then thefunction V (t xPξ ) defined above is the unique classical solution of the followingnonlocal PDE of mean-field type (see Theorem 72)
parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )(16)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
= 0
V (T xPξ ) = (xPξ )
MEAN-FIELD SDES AND ASSOCIATED PDES 827
with (t xPξ ) isin [0 T ]timesRd timesP2(R
d) We see in particular that in contrast to theclassical case the derivative partμV (t xPξ y) and as a second-order derivative thederivative of partμV (t xPξ y) with respect to y are involved
Mean-field SDEs also known as McKeanndashVlasov equations were discussedthe first time by Kac [13 14] in the frame of his study of the Boltzman equa-tion for the particle density in diluted monatomic gases as well as in that of thestochastic toy model for the Vlasov kinetic equation of plasma A by now classi-cal method of solving mean-field SDEs by approximation consists in the use ofso-called N -particle systems with weak interaction formed by N equations drivenby independent Brownian motions The convergence of this system to the mean-field SDE is called in the literature propagation of chaos for the McKeanndashVlasovequation
The pioneering works by Kac and in the aftermath by other authors have at-tracted a lot of researchers interested in the study of the chaos propagation andthe limit equations in different frameworks for getting an impression we refer thereader for instance to [1 3 10ndash12 15 18ndash20] and [21] as well as the referencestherein With the pioneering works on mean-field stochastic differential games byLasry and Lions (we refer to [16] and the papers cited therein but also to [6]) newimpulses and new applications for mean-field problems were given So recentlyBuckdahn Djehiche Li and Peng [4] studied a special mean field problem by apurely stochastic approach and deduced a new kind of backward SDE (BSDE)which they called mean-field BSDE they showed that the BSDE can be obtainedby an approximation involving N -particle systems with weak interaction Theycompleted these studies of the approximation with associating a kind of centrallimit theorem for the approximating systems and obtained as limit some forward-backward SDE of mean-field type which is not only governed by a Brownianmotion but also by an independent Gaussian field In [5] deepening the investi-gation of mean-field SDEs and associated mean-field BSDEs Buckdahn Li andPeng generalized their previous results on mean-field BSDEs and in a ldquoMarko-vianrdquo framework in which the initial data (t x) were frozen in the law variable ofthe coefficients they investigated the associated nonlocal PDE However our ob-jective has been to overcome this partial freezing of initial data in the mean-fieldSDE and to study the associated PDE and this is done in our present manuscriptOur approach is highly inspired by the courses given by Lions [17] at Collegravege deFrance (redacted by Cardaliaguet [6]) and by recent works of Bensoussan FrehseYam [2] and Carmona and Delarue who directly inspired by the works of LasryLions [16] and the courses of Lions [17] translated his rather analytical approachinto a stochastic one let us cite [8 9] and the references indicated therein
Let us describe the organization of our manuscript and link this description withthe explanation of the novelty in our approach
In Section 2 ldquoPreliminariesrdquo we introduce the framework of our study Partic-ular attention is paid to a recall of Lionsrsquo definition of the derivative of a function
828 BUCKDAHN LI PENG AND RAINER
defined over the space of probability measures over Rd with finite second mo-
ment On the basis of this notion of first-order derivatives we introduce in exactlythe same spirit the second-order derivatives of such a function The first- and thesecond-order differentiability of a function with respect to a probability measureallows in the following to derive a second-order Taylor expansion (Lemma 21)which is new and turns out to be crucial in our approach We give an example(Example 23) which shows that in general even in the most regular cases thefunctions lifted from the space of probability measures with finite second momentto the space of square integrable random variables are not twice Freacutechet differen-tiable on L2 but well twice differentiable with respect to the probability measureTo the best of our knowledge the passage to higher order derivatives with respectto the measure the second-order Taylor expansion with respect to the measure andExample 23 are new They constitute the crucial paves in our approach in par-ticular also for the derivation of the Itocirc formula for mean-field Itocirc processes (seeTheorem 71 in Section 7)
In Section 3 we introduce our mean-field SDE with our standard assumptionson its coefficients [their twice fold differentiability with respect to (xμ) withbounded Lipschitz derivatives of first and second order] and we study useful prop-erties of the mean-field SDE A crucial step in our approach constitutes the splittingof mean-field SDE (11) into (13) and (14)mdasha system of mean-field SDEs whichunlike (11) generates a flow The flow concerns the couple formed by the stateand by the law of the process Such a splitting for the study of mean-field SDEs isnovel
A central property studied in Section 4 is the differentiability of its solutionprocess XtxPξ with respect to the probability law Pξ The identification of thederivative of XtxPξ with respect to the probability law and the equation satisfiedby it are new to the best of our knowledge The investigations of Section 4 are com-pleted by Section 5 which is devoted to the study of the second-order derivativesof XtxPξ and so namely for that with respect to the probability law The first- andthe second-order derivatives of XtxPξ are characterized as the unique solution ofassociated SDEs which on their part allow to get estimates for the derivatives oforder 1 and 2 of XtxPξ The results obtained for the process XtxPξ and so alsofor Xtξ in the Sections 3ndash5 are used in Section 6 for the proof of the regularity ofthe value function V (t xPξ ) Finally Section 7 is devoted to a novel Itocirc formulaassociated with mean-field problems and it gives our main result Theorem 72stating that our value function V is the unique classical solution of the PDE (16)of mean-field type given above
2 Preliminaries Let us begin with introducing some notation and con-cepts which we will need in our further computations We shall in particularintroduce the notion of differentiability of a function f defined over the spaceP2(R
d) of all probability measures μ over (RdB(Rd)) with finite second mo-ment
intRd |x|2μ(dx) lt infin [B(Rd) denotes the Borel σ -field over Rd ] The space
MEAN-FIELD SDES AND ASSOCIATED PDES 829
P2(R2d) is endowed with the 2-Wasserstein metric
W2(μ ν) = inf(int
RdtimesRd|x minus y|2ρ(dx dy)
)12
ρ isin P2(R
2d)(21)
with ρ(middot timesR
d) = μρ(R
d times middot) = ν
μ ν isin P2
(R
d)
Among the different notions of differentiability of a function f defined overP2(R
d) we adopt for our approach that introduced by Lions [17] in his lectures atCollegravege de France in Paris and revised in the notes by Cardaliaguet [6] we referthe reader also for instance to Carmona and Delarue [9] Let us consider a proba-bility space (FP ) which is ldquorich enoughrdquo (the precise space we will work withwill be introduced later) ldquoRich enoughrdquo means that for every μ isin P2(R
d) thereis a random variable ϑ isin L2(FRd)(= L2(FP Rd)) such that Pϑ = μ It iswell known that the probability space ([01]B([01]) dx) has this property
Identifying the random variables in L2(FRd) which coincide P -as we canregard L2(FRd) as a Hilbert space with inner product (ξ η)L2 = E[ξ middot η] ξ η isinL2(FRd) and norm |ξ |L2 = (ξ ξ)
12L2 Recall that due to the definition made
by Lions [6] (see Cardaliaguet [7]) a function f P2(Rd) rarr R is said to be dif-
ferentiable in μ isin P2(Rd) if for f (ϑ) = f (Pϑ)ϑ isin L2(FRd) there is some
ϑ0 isin L2(FRd) with Pϑ0 = μ such that the function f L2(FRd) rarr R is dif-ferentiable (in Freacutechet sense) in ϑ0 that is there exists a linear continuous mappingDf (ϑ0) L2(FRd) rarrR (Df (ϑ0) isin L(L2(FRd)R)) such that
f (ϑ0 + η) minus f (ϑ0) = Df (ϑ0)(η) + o(|η|L2
)(22)
with |η|L2 rarr 0 for η isin L2(FRd) Since Df (ϑ0) isin L(L2(FRd)R) Rieszrsquo rep-resentation theorem yields the existence of a (P -as) unique random variable θ0 isinL2(FRd) such that Df (ϑ0)(η) = (θ0 η)L2 = E[θ0η] for all η isin L2(FRd) In[17] (see also [6]) it has been proved that there is a Borel function h0 Rd rarr R
d
such that θ0 = h0(ϑ0)P -as The function h0 only depends on the law Pϑ0 but noton ϑ0 itself Taking into account the definition of f this allows to write
f (Pϑ) minus f (Pϑ0) = E[h0(ϑ0) middot (ϑ minus ϑ0)
] + o(|ϑ minus ϑ0|L2
)(23)
ϑ isin L2(FRd)We call partμf (Pϑ0 y) = h0(y) y isin R
d the derivative of f P2(Rd) rarr R at
Pϑ0 Note that partμf (Pϑ0 y) is only Pϑ0(dy)-ae uniquely determinedHowever in our approach we have to consider functions f P2(R
d) rarrR whichare differentiable in all elements of P2(R
d) In order to simplify the argumentwe suppose that f L2(FRd) rarr R is Freacutechet differential over the whole spaceL2(FRd) This corresponds to a large class of important examples In this casewe have the derivative partμf (Pϑ y) defined Pϑ(dy)-ae for all ϑ isin L2(FRd)In Lemma 32 [9] it is shown that if furthermore the Freacutechet derivative Df L2(FRd) rarr L(L2(FRd)R) is Lipschitz continuous (with a Lipschitz constant
830 BUCKDAHN LI PENG AND RAINER
K isin R+) then there is for all ϑ isin L2(FRd) a Pϑ -version of partμf (Pϑ middot) Rd rarrR
d such that∣∣partμf (Pϑ y) minus partμf(Pϑ yprime)∣∣ le K
∣∣y minus yprime∣∣ for all y yprime isin Rd
This motivates us to make the following definition
DEFINITION 21 We say that f isin C11b (P2(R
d)) [continuously differentiableover P2(R
d) with Lipschitz-continuous bounded derivative] if there exists for allϑ isin L2(FRd) a Pϑ -modification of partμf (Pϑ middot) again denoted by partμf (Pϑ middot)such that partμf P2(R
d) timesRd rarr R
d is bounded and Lipschitz continuous that isthere is some real constant C such that
(i) |partμf (μx)| le Cμ isin P2(Rd) x isin R
d (ii) |partμf (μx) minus partμf (μprime xprime)| le C(W2(μμprime) + |x minus xprime|)μμprime isin P2(R
d) xxprime isin R
d
we consider this function partμf as the derivative of f
REMARK 21 Let us point out that if f isin C11b (P2(R
d)) the version ofpartμf (Pϑ middot) ϑ isin L2(FRd) indicated in Definition 21 is unique Indeed givenϑ isin L2(FRd) let θ be a d-dimensional vector of independent standard nor-mally distributed random variables which are independent of ϑ Then sincepartμf (Pϑ+εθ ϑ + εθ) is only P -as defined partμf (Pϑ+εθ y) is determined onlyPϑ+εθ (dy)-as Observing that the random variable ϑ +εθ possesses a strictly pos-itive density on R
d it follows that Pϑ+εθ and the Lebesgue measure over Rd areequivalent Consequently partμf (Pϑ+εθ y) is defined dy-ae on R
d From the Lip-schitz continuity (ii) of partμf in Definition 21 it then follows that partμf (Pϑ+εθ y)
is defined for all y isin Rd and taking the limit 0 lt ε darr 0 yields that partμf (Pϑ y) is
uniquely defined for all y isin Rd
Given now f isin C11b (P2(R
d)) for fixed y isin Rd the question of the differen-
tiability of its components (partμf )j (middot y) P2(Rd) rarr R1 le j le d raises and
it can be discussed in the same way as the first-order derivative partμf above If(partμf )j (middot y) P2(R
d) rarr R belongs to C11b (P2(R
d)) we have that its derivativepartμ((partμf )j (middot y))(middot middot) P2(R
d)timesRd rarrR
d is a Lipschitz-continuous function forevery y isin R
d Then
part2μf (μx y) = (
partμ
((partμf )j (middot y)
)(μ x)
)1lejled
(24)(μ x y) isin P2
(R
d) timesR
d timesRd
defines a function part2μf P2(R
d) timesRd timesR
d rarrRd otimesR
d
DEFINITION 22 We say that f isin C21b (P2(R
d)) if f isin C11b (P2(R
d)) and
MEAN-FIELD SDES AND ASSOCIATED PDES 831
(i) (partμf )j (middot y) isin C11b (P2(R
d)) for all y isin Rd1 le j le d and part2
μf P2(R
d) timesRd timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous(ii) partμf (μ middot) Rd rarr R
d is differentiable for every μ isin P2(Rd) and its
derivative partypartμf P2(Rd)timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous
Adopting the above introduced notation we consider a functionf isin C
21b (P2(R
d)) and discuss its second-order Taylor expansion For this endwe have still to introduce some notation Let ( F P ) be a copy of the prob-ability space (FP ) For any random variable (of arbitrary dimension) ϑ
over (FP ) we denote by ϑ a copy (of the same law as ϑ but definedover ( F P ) Pϑ = Pϑ The expectation E[middot] = int
(middot) dP acts only over thevariables endowed with a tilde This can be made rigorous by working withthe product space (FP ) otimes ( F P ) = (FP ) otimes (FP ) and puttingϑ(ωω) = ϑ(ω) (ωω) isin times = times for ϑ random variable defined over(FP )) Of course this formalism can be easily extended from random vari-ables to stochastic processes
With the above notation and writing a otimes b = (aibj )1leijled for a = (ai)1leiled
b = (bj )1lejled isin Rd we can state now the following result
LEMMA 21 Let f isin C21b (P2(R
d)) Then for any given ϑ0 isin L2(FRd) wehave the following second-order expansion
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](25)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R(PϑPϑ0)
ϑ isin L2(FRd
)
where η = ϑ minus ϑ0 and for all ϑ isin L2(FRd) the remainder R(PϑPϑ0) satisfiesthe estimate ∣∣R(PϑPϑ0)
∣∣ le C((
E[|ϑ minus ϑ0|2])32 + E
[|ϑ minus ϑ0|3])(26)
le CE[|ϑ minus ϑ0|3]
The constant C isin R+ only depends on the Lipschitz constants of part2μf and partypartμf
We observe that the above second-order expansion does not constitute a second-order Taylor expansion for the associated function f L2(FRd) rarr R since theremainder term E[|ϑ minus ϑ0|3 and |ϑ minus ϑ0|2] is not of order o(|ϑ minus ϑ0|2L2) Indeed asthe following example shows in general we only have E[|ϑ minusϑ0|3 and |ϑ minusϑ0|2] =O(|ϑ minus ϑ0|2L2)
832 BUCKDAHN LI PENG AND RAINER
EXAMPLE 21 Let ϑ = IA with A isin F such that P(A) rarr 0 as rarr infin
Then ϑ rarr 0 in L2( rarr infin) and E[|ϑ|3 and |ϑ|2] = P(A) = E[ϑ2 ] = |ϑ|2L2 rarr
0 ( rarr infin)
If we had in Lemma 21 a remainder R(PϑPϑ0) = o(|ϑ minus ϑ0|2L2) this wouldsuggest that f (ζ ) = f (Pζ ) is twice Freacutechet differentiable at ϑ0 But however asthe following Example 23 shows even in easiest cases f is not twice Freacutechetdifferentiable
However for our purposes the above expansion is fine
PROOF OF LEMMA 21 Let ϑ0 isin L2(FRd) Then for all ϑ isin L2(FRd)putting η = ϑ minus ϑ0 and using the fact that f isin C
21b (P2(R
d)) we have
f (Pϑ) minus f (Pϑ0) =int 1
0
d
dλf (Pϑ0+λη) dλ
(27)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
Let us now compute ddλ
partμf (Pϑ0+ληϑ0 +λη) Since f isin C21b (P2(R
d)) the liftedfunction partμf (ξ y) = partμf (Pξ y) ξ isin L2(FRd) is Freacutechet differentiable in ξ and
d
dλpartμf (Pϑ0+λη y) = d
dλpartμf (ϑ0 + λη y)
(28)= E
[part2μf (Pϑ0+ληϑ0 + λη y) middot η]
λ isin R y isin Rd Then choosing an independent copy (ϑ ϑ0) of (ϑϑ0) defined
over ( F P ) (ie in particular P(ϑϑ0)= P(ϑϑ0)) we have for η = ϑ minus ϑ0
d
dλpartμf (Pϑ0+λη y) = E
[part2μf (Pϑ0+λη ϑ0 + λη y) middot η]
(29)λ isin R y isin R
d
Consequently
d
dλpartμf (Pϑ0+ληϑ0 + λη) = E
[part2μf (Pϑ0+λη ϑ0 + ληϑ0 + λη) middot η]
(210)+ partypartμf (Pϑ0+ληϑ0 + λη) middot η
Thus
f (Pϑ) minus f (Pϑ0)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
MEAN-FIELD SDES AND ASSOCIATED PDES 833
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[d
dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η
]dρ dλ(211)
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[tr
(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η
)]dρ dλ
+int 1
0
int λ
0E
[E
[tr
(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η
)]]dρ dλ
From this latter relation we get
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](212)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R1(PϑPϑ0) + R2(PϑPϑ0)
with the remainders
R1(PϑPϑ0) =int 1
0
int λ
0E
[E
[tr
((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)
(213)minus part2
μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η
)]]dρ dλ
and
R2(PϑPϑ0) =int 1
0
int λ
0E
[tr
((partypartμf (Pϑ0+ρηϑ0 + ρη)
(214)minus partypartμf (Pϑ0 ϑ0)
) middot η otimes η)]
dρ dλ
Finally from the boundedness and the Lipschitz continuity of the functions part2μf
and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf
|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement
Let us finish our preliminary discussion with two illustrating examples
EXAMPLE 22 Given two twice continuously differentiable functions h R
d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and
f (ϑ0 + η) minus f (ϑ0) =int 1
0gprime(E[
h(ϑ0 + sη)])
E[hprime(ϑ0 + sη)η
]ds
= gprime(E[h(ϑ0)
])E
[hprime(ϑ0)η
] + o(|η|L2
)= E
[gprime(E[
h(ϑ0)])
hprime(ϑ0)η] + o
(|η|L2)
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
MEAN-FIELD SDES AND ASSOCIATED PDES 827
with (t xPξ ) isin [0 T ]timesRd timesP2(R
d) We see in particular that in contrast to theclassical case the derivative partμV (t xPξ y) and as a second-order derivative thederivative of partμV (t xPξ y) with respect to y are involved
Mean-field SDEs also known as McKeanndashVlasov equations were discussedthe first time by Kac [13 14] in the frame of his study of the Boltzman equa-tion for the particle density in diluted monatomic gases as well as in that of thestochastic toy model for the Vlasov kinetic equation of plasma A by now classi-cal method of solving mean-field SDEs by approximation consists in the use ofso-called N -particle systems with weak interaction formed by N equations drivenby independent Brownian motions The convergence of this system to the mean-field SDE is called in the literature propagation of chaos for the McKeanndashVlasovequation
The pioneering works by Kac and in the aftermath by other authors have at-tracted a lot of researchers interested in the study of the chaos propagation andthe limit equations in different frameworks for getting an impression we refer thereader for instance to [1 3 10ndash12 15 18ndash20] and [21] as well as the referencestherein With the pioneering works on mean-field stochastic differential games byLasry and Lions (we refer to [16] and the papers cited therein but also to [6]) newimpulses and new applications for mean-field problems were given So recentlyBuckdahn Djehiche Li and Peng [4] studied a special mean field problem by apurely stochastic approach and deduced a new kind of backward SDE (BSDE)which they called mean-field BSDE they showed that the BSDE can be obtainedby an approximation involving N -particle systems with weak interaction Theycompleted these studies of the approximation with associating a kind of centrallimit theorem for the approximating systems and obtained as limit some forward-backward SDE of mean-field type which is not only governed by a Brownianmotion but also by an independent Gaussian field In [5] deepening the investi-gation of mean-field SDEs and associated mean-field BSDEs Buckdahn Li andPeng generalized their previous results on mean-field BSDEs and in a ldquoMarko-vianrdquo framework in which the initial data (t x) were frozen in the law variable ofthe coefficients they investigated the associated nonlocal PDE However our ob-jective has been to overcome this partial freezing of initial data in the mean-fieldSDE and to study the associated PDE and this is done in our present manuscriptOur approach is highly inspired by the courses given by Lions [17] at Collegravege deFrance (redacted by Cardaliaguet [6]) and by recent works of Bensoussan FrehseYam [2] and Carmona and Delarue who directly inspired by the works of LasryLions [16] and the courses of Lions [17] translated his rather analytical approachinto a stochastic one let us cite [8 9] and the references indicated therein
Let us describe the organization of our manuscript and link this description withthe explanation of the novelty in our approach
In Section 2 ldquoPreliminariesrdquo we introduce the framework of our study Partic-ular attention is paid to a recall of Lionsrsquo definition of the derivative of a function
828 BUCKDAHN LI PENG AND RAINER
defined over the space of probability measures over Rd with finite second mo-
ment On the basis of this notion of first-order derivatives we introduce in exactlythe same spirit the second-order derivatives of such a function The first- and thesecond-order differentiability of a function with respect to a probability measureallows in the following to derive a second-order Taylor expansion (Lemma 21)which is new and turns out to be crucial in our approach We give an example(Example 23) which shows that in general even in the most regular cases thefunctions lifted from the space of probability measures with finite second momentto the space of square integrable random variables are not twice Freacutechet differen-tiable on L2 but well twice differentiable with respect to the probability measureTo the best of our knowledge the passage to higher order derivatives with respectto the measure the second-order Taylor expansion with respect to the measure andExample 23 are new They constitute the crucial paves in our approach in par-ticular also for the derivation of the Itocirc formula for mean-field Itocirc processes (seeTheorem 71 in Section 7)
In Section 3 we introduce our mean-field SDE with our standard assumptionson its coefficients [their twice fold differentiability with respect to (xμ) withbounded Lipschitz derivatives of first and second order] and we study useful prop-erties of the mean-field SDE A crucial step in our approach constitutes the splittingof mean-field SDE (11) into (13) and (14)mdasha system of mean-field SDEs whichunlike (11) generates a flow The flow concerns the couple formed by the stateand by the law of the process Such a splitting for the study of mean-field SDEs isnovel
A central property studied in Section 4 is the differentiability of its solutionprocess XtxPξ with respect to the probability law Pξ The identification of thederivative of XtxPξ with respect to the probability law and the equation satisfiedby it are new to the best of our knowledge The investigations of Section 4 are com-pleted by Section 5 which is devoted to the study of the second-order derivativesof XtxPξ and so namely for that with respect to the probability law The first- andthe second-order derivatives of XtxPξ are characterized as the unique solution ofassociated SDEs which on their part allow to get estimates for the derivatives oforder 1 and 2 of XtxPξ The results obtained for the process XtxPξ and so alsofor Xtξ in the Sections 3ndash5 are used in Section 6 for the proof of the regularity ofthe value function V (t xPξ ) Finally Section 7 is devoted to a novel Itocirc formulaassociated with mean-field problems and it gives our main result Theorem 72stating that our value function V is the unique classical solution of the PDE (16)of mean-field type given above
2 Preliminaries Let us begin with introducing some notation and con-cepts which we will need in our further computations We shall in particularintroduce the notion of differentiability of a function f defined over the spaceP2(R
d) of all probability measures μ over (RdB(Rd)) with finite second mo-ment
intRd |x|2μ(dx) lt infin [B(Rd) denotes the Borel σ -field over Rd ] The space
MEAN-FIELD SDES AND ASSOCIATED PDES 829
P2(R2d) is endowed with the 2-Wasserstein metric
W2(μ ν) = inf(int
RdtimesRd|x minus y|2ρ(dx dy)
)12
ρ isin P2(R
2d)(21)
with ρ(middot timesR
d) = μρ(R
d times middot) = ν
μ ν isin P2
(R
d)
Among the different notions of differentiability of a function f defined overP2(R
d) we adopt for our approach that introduced by Lions [17] in his lectures atCollegravege de France in Paris and revised in the notes by Cardaliaguet [6] we referthe reader also for instance to Carmona and Delarue [9] Let us consider a proba-bility space (FP ) which is ldquorich enoughrdquo (the precise space we will work withwill be introduced later) ldquoRich enoughrdquo means that for every μ isin P2(R
d) thereis a random variable ϑ isin L2(FRd)(= L2(FP Rd)) such that Pϑ = μ It iswell known that the probability space ([01]B([01]) dx) has this property
Identifying the random variables in L2(FRd) which coincide P -as we canregard L2(FRd) as a Hilbert space with inner product (ξ η)L2 = E[ξ middot η] ξ η isinL2(FRd) and norm |ξ |L2 = (ξ ξ)
12L2 Recall that due to the definition made
by Lions [6] (see Cardaliaguet [7]) a function f P2(Rd) rarr R is said to be dif-
ferentiable in μ isin P2(Rd) if for f (ϑ) = f (Pϑ)ϑ isin L2(FRd) there is some
ϑ0 isin L2(FRd) with Pϑ0 = μ such that the function f L2(FRd) rarr R is dif-ferentiable (in Freacutechet sense) in ϑ0 that is there exists a linear continuous mappingDf (ϑ0) L2(FRd) rarrR (Df (ϑ0) isin L(L2(FRd)R)) such that
f (ϑ0 + η) minus f (ϑ0) = Df (ϑ0)(η) + o(|η|L2
)(22)
with |η|L2 rarr 0 for η isin L2(FRd) Since Df (ϑ0) isin L(L2(FRd)R) Rieszrsquo rep-resentation theorem yields the existence of a (P -as) unique random variable θ0 isinL2(FRd) such that Df (ϑ0)(η) = (θ0 η)L2 = E[θ0η] for all η isin L2(FRd) In[17] (see also [6]) it has been proved that there is a Borel function h0 Rd rarr R
d
such that θ0 = h0(ϑ0)P -as The function h0 only depends on the law Pϑ0 but noton ϑ0 itself Taking into account the definition of f this allows to write
f (Pϑ) minus f (Pϑ0) = E[h0(ϑ0) middot (ϑ minus ϑ0)
] + o(|ϑ minus ϑ0|L2
)(23)
ϑ isin L2(FRd)We call partμf (Pϑ0 y) = h0(y) y isin R
d the derivative of f P2(Rd) rarr R at
Pϑ0 Note that partμf (Pϑ0 y) is only Pϑ0(dy)-ae uniquely determinedHowever in our approach we have to consider functions f P2(R
d) rarrR whichare differentiable in all elements of P2(R
d) In order to simplify the argumentwe suppose that f L2(FRd) rarr R is Freacutechet differential over the whole spaceL2(FRd) This corresponds to a large class of important examples In this casewe have the derivative partμf (Pϑ y) defined Pϑ(dy)-ae for all ϑ isin L2(FRd)In Lemma 32 [9] it is shown that if furthermore the Freacutechet derivative Df L2(FRd) rarr L(L2(FRd)R) is Lipschitz continuous (with a Lipschitz constant
830 BUCKDAHN LI PENG AND RAINER
K isin R+) then there is for all ϑ isin L2(FRd) a Pϑ -version of partμf (Pϑ middot) Rd rarrR
d such that∣∣partμf (Pϑ y) minus partμf(Pϑ yprime)∣∣ le K
∣∣y minus yprime∣∣ for all y yprime isin Rd
This motivates us to make the following definition
DEFINITION 21 We say that f isin C11b (P2(R
d)) [continuously differentiableover P2(R
d) with Lipschitz-continuous bounded derivative] if there exists for allϑ isin L2(FRd) a Pϑ -modification of partμf (Pϑ middot) again denoted by partμf (Pϑ middot)such that partμf P2(R
d) timesRd rarr R
d is bounded and Lipschitz continuous that isthere is some real constant C such that
(i) |partμf (μx)| le Cμ isin P2(Rd) x isin R
d (ii) |partμf (μx) minus partμf (μprime xprime)| le C(W2(μμprime) + |x minus xprime|)μμprime isin P2(R
d) xxprime isin R
d
we consider this function partμf as the derivative of f
REMARK 21 Let us point out that if f isin C11b (P2(R
d)) the version ofpartμf (Pϑ middot) ϑ isin L2(FRd) indicated in Definition 21 is unique Indeed givenϑ isin L2(FRd) let θ be a d-dimensional vector of independent standard nor-mally distributed random variables which are independent of ϑ Then sincepartμf (Pϑ+εθ ϑ + εθ) is only P -as defined partμf (Pϑ+εθ y) is determined onlyPϑ+εθ (dy)-as Observing that the random variable ϑ +εθ possesses a strictly pos-itive density on R
d it follows that Pϑ+εθ and the Lebesgue measure over Rd areequivalent Consequently partμf (Pϑ+εθ y) is defined dy-ae on R
d From the Lip-schitz continuity (ii) of partμf in Definition 21 it then follows that partμf (Pϑ+εθ y)
is defined for all y isin Rd and taking the limit 0 lt ε darr 0 yields that partμf (Pϑ y) is
uniquely defined for all y isin Rd
Given now f isin C11b (P2(R
d)) for fixed y isin Rd the question of the differen-
tiability of its components (partμf )j (middot y) P2(Rd) rarr R1 le j le d raises and
it can be discussed in the same way as the first-order derivative partμf above If(partμf )j (middot y) P2(R
d) rarr R belongs to C11b (P2(R
d)) we have that its derivativepartμ((partμf )j (middot y))(middot middot) P2(R
d)timesRd rarrR
d is a Lipschitz-continuous function forevery y isin R
d Then
part2μf (μx y) = (
partμ
((partμf )j (middot y)
)(μ x)
)1lejled
(24)(μ x y) isin P2
(R
d) timesR
d timesRd
defines a function part2μf P2(R
d) timesRd timesR
d rarrRd otimesR
d
DEFINITION 22 We say that f isin C21b (P2(R
d)) if f isin C11b (P2(R
d)) and
MEAN-FIELD SDES AND ASSOCIATED PDES 831
(i) (partμf )j (middot y) isin C11b (P2(R
d)) for all y isin Rd1 le j le d and part2
μf P2(R
d) timesRd timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous(ii) partμf (μ middot) Rd rarr R
d is differentiable for every μ isin P2(Rd) and its
derivative partypartμf P2(Rd)timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous
Adopting the above introduced notation we consider a functionf isin C
21b (P2(R
d)) and discuss its second-order Taylor expansion For this endwe have still to introduce some notation Let ( F P ) be a copy of the prob-ability space (FP ) For any random variable (of arbitrary dimension) ϑ
over (FP ) we denote by ϑ a copy (of the same law as ϑ but definedover ( F P ) Pϑ = Pϑ The expectation E[middot] = int
(middot) dP acts only over thevariables endowed with a tilde This can be made rigorous by working withthe product space (FP ) otimes ( F P ) = (FP ) otimes (FP ) and puttingϑ(ωω) = ϑ(ω) (ωω) isin times = times for ϑ random variable defined over(FP )) Of course this formalism can be easily extended from random vari-ables to stochastic processes
With the above notation and writing a otimes b = (aibj )1leijled for a = (ai)1leiled
b = (bj )1lejled isin Rd we can state now the following result
LEMMA 21 Let f isin C21b (P2(R
d)) Then for any given ϑ0 isin L2(FRd) wehave the following second-order expansion
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](25)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R(PϑPϑ0)
ϑ isin L2(FRd
)
where η = ϑ minus ϑ0 and for all ϑ isin L2(FRd) the remainder R(PϑPϑ0) satisfiesthe estimate ∣∣R(PϑPϑ0)
∣∣ le C((
E[|ϑ minus ϑ0|2])32 + E
[|ϑ minus ϑ0|3])(26)
le CE[|ϑ minus ϑ0|3]
The constant C isin R+ only depends on the Lipschitz constants of part2μf and partypartμf
We observe that the above second-order expansion does not constitute a second-order Taylor expansion for the associated function f L2(FRd) rarr R since theremainder term E[|ϑ minus ϑ0|3 and |ϑ minus ϑ0|2] is not of order o(|ϑ minus ϑ0|2L2) Indeed asthe following example shows in general we only have E[|ϑ minusϑ0|3 and |ϑ minusϑ0|2] =O(|ϑ minus ϑ0|2L2)
832 BUCKDAHN LI PENG AND RAINER
EXAMPLE 21 Let ϑ = IA with A isin F such that P(A) rarr 0 as rarr infin
Then ϑ rarr 0 in L2( rarr infin) and E[|ϑ|3 and |ϑ|2] = P(A) = E[ϑ2 ] = |ϑ|2L2 rarr
0 ( rarr infin)
If we had in Lemma 21 a remainder R(PϑPϑ0) = o(|ϑ minus ϑ0|2L2) this wouldsuggest that f (ζ ) = f (Pζ ) is twice Freacutechet differentiable at ϑ0 But however asthe following Example 23 shows even in easiest cases f is not twice Freacutechetdifferentiable
However for our purposes the above expansion is fine
PROOF OF LEMMA 21 Let ϑ0 isin L2(FRd) Then for all ϑ isin L2(FRd)putting η = ϑ minus ϑ0 and using the fact that f isin C
21b (P2(R
d)) we have
f (Pϑ) minus f (Pϑ0) =int 1
0
d
dλf (Pϑ0+λη) dλ
(27)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
Let us now compute ddλ
partμf (Pϑ0+ληϑ0 +λη) Since f isin C21b (P2(R
d)) the liftedfunction partμf (ξ y) = partμf (Pξ y) ξ isin L2(FRd) is Freacutechet differentiable in ξ and
d
dλpartμf (Pϑ0+λη y) = d
dλpartμf (ϑ0 + λη y)
(28)= E
[part2μf (Pϑ0+ληϑ0 + λη y) middot η]
λ isin R y isin Rd Then choosing an independent copy (ϑ ϑ0) of (ϑϑ0) defined
over ( F P ) (ie in particular P(ϑϑ0)= P(ϑϑ0)) we have for η = ϑ minus ϑ0
d
dλpartμf (Pϑ0+λη y) = E
[part2μf (Pϑ0+λη ϑ0 + λη y) middot η]
(29)λ isin R y isin R
d
Consequently
d
dλpartμf (Pϑ0+ληϑ0 + λη) = E
[part2μf (Pϑ0+λη ϑ0 + ληϑ0 + λη) middot η]
(210)+ partypartμf (Pϑ0+ληϑ0 + λη) middot η
Thus
f (Pϑ) minus f (Pϑ0)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
MEAN-FIELD SDES AND ASSOCIATED PDES 833
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[d
dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η
]dρ dλ(211)
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[tr
(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η
)]dρ dλ
+int 1
0
int λ
0E
[E
[tr
(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η
)]]dρ dλ
From this latter relation we get
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](212)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R1(PϑPϑ0) + R2(PϑPϑ0)
with the remainders
R1(PϑPϑ0) =int 1
0
int λ
0E
[E
[tr
((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)
(213)minus part2
μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η
)]]dρ dλ
and
R2(PϑPϑ0) =int 1
0
int λ
0E
[tr
((partypartμf (Pϑ0+ρηϑ0 + ρη)
(214)minus partypartμf (Pϑ0 ϑ0)
) middot η otimes η)]
dρ dλ
Finally from the boundedness and the Lipschitz continuity of the functions part2μf
and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf
|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement
Let us finish our preliminary discussion with two illustrating examples
EXAMPLE 22 Given two twice continuously differentiable functions h R
d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and
f (ϑ0 + η) minus f (ϑ0) =int 1
0gprime(E[
h(ϑ0 + sη)])
E[hprime(ϑ0 + sη)η
]ds
= gprime(E[h(ϑ0)
])E
[hprime(ϑ0)η
] + o(|η|L2
)= E
[gprime(E[
h(ϑ0)])
hprime(ϑ0)η] + o
(|η|L2)
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
828 BUCKDAHN LI PENG AND RAINER
defined over the space of probability measures over Rd with finite second mo-
ment On the basis of this notion of first-order derivatives we introduce in exactlythe same spirit the second-order derivatives of such a function The first- and thesecond-order differentiability of a function with respect to a probability measureallows in the following to derive a second-order Taylor expansion (Lemma 21)which is new and turns out to be crucial in our approach We give an example(Example 23) which shows that in general even in the most regular cases thefunctions lifted from the space of probability measures with finite second momentto the space of square integrable random variables are not twice Freacutechet differen-tiable on L2 but well twice differentiable with respect to the probability measureTo the best of our knowledge the passage to higher order derivatives with respectto the measure the second-order Taylor expansion with respect to the measure andExample 23 are new They constitute the crucial paves in our approach in par-ticular also for the derivation of the Itocirc formula for mean-field Itocirc processes (seeTheorem 71 in Section 7)
In Section 3 we introduce our mean-field SDE with our standard assumptionson its coefficients [their twice fold differentiability with respect to (xμ) withbounded Lipschitz derivatives of first and second order] and we study useful prop-erties of the mean-field SDE A crucial step in our approach constitutes the splittingof mean-field SDE (11) into (13) and (14)mdasha system of mean-field SDEs whichunlike (11) generates a flow The flow concerns the couple formed by the stateand by the law of the process Such a splitting for the study of mean-field SDEs isnovel
A central property studied in Section 4 is the differentiability of its solutionprocess XtxPξ with respect to the probability law Pξ The identification of thederivative of XtxPξ with respect to the probability law and the equation satisfiedby it are new to the best of our knowledge The investigations of Section 4 are com-pleted by Section 5 which is devoted to the study of the second-order derivativesof XtxPξ and so namely for that with respect to the probability law The first- andthe second-order derivatives of XtxPξ are characterized as the unique solution ofassociated SDEs which on their part allow to get estimates for the derivatives oforder 1 and 2 of XtxPξ The results obtained for the process XtxPξ and so alsofor Xtξ in the Sections 3ndash5 are used in Section 6 for the proof of the regularity ofthe value function V (t xPξ ) Finally Section 7 is devoted to a novel Itocirc formulaassociated with mean-field problems and it gives our main result Theorem 72stating that our value function V is the unique classical solution of the PDE (16)of mean-field type given above
2 Preliminaries Let us begin with introducing some notation and con-cepts which we will need in our further computations We shall in particularintroduce the notion of differentiability of a function f defined over the spaceP2(R
d) of all probability measures μ over (RdB(Rd)) with finite second mo-ment
intRd |x|2μ(dx) lt infin [B(Rd) denotes the Borel σ -field over Rd ] The space
MEAN-FIELD SDES AND ASSOCIATED PDES 829
P2(R2d) is endowed with the 2-Wasserstein metric
W2(μ ν) = inf(int
RdtimesRd|x minus y|2ρ(dx dy)
)12
ρ isin P2(R
2d)(21)
with ρ(middot timesR
d) = μρ(R
d times middot) = ν
μ ν isin P2
(R
d)
Among the different notions of differentiability of a function f defined overP2(R
d) we adopt for our approach that introduced by Lions [17] in his lectures atCollegravege de France in Paris and revised in the notes by Cardaliaguet [6] we referthe reader also for instance to Carmona and Delarue [9] Let us consider a proba-bility space (FP ) which is ldquorich enoughrdquo (the precise space we will work withwill be introduced later) ldquoRich enoughrdquo means that for every μ isin P2(R
d) thereis a random variable ϑ isin L2(FRd)(= L2(FP Rd)) such that Pϑ = μ It iswell known that the probability space ([01]B([01]) dx) has this property
Identifying the random variables in L2(FRd) which coincide P -as we canregard L2(FRd) as a Hilbert space with inner product (ξ η)L2 = E[ξ middot η] ξ η isinL2(FRd) and norm |ξ |L2 = (ξ ξ)
12L2 Recall that due to the definition made
by Lions [6] (see Cardaliaguet [7]) a function f P2(Rd) rarr R is said to be dif-
ferentiable in μ isin P2(Rd) if for f (ϑ) = f (Pϑ)ϑ isin L2(FRd) there is some
ϑ0 isin L2(FRd) with Pϑ0 = μ such that the function f L2(FRd) rarr R is dif-ferentiable (in Freacutechet sense) in ϑ0 that is there exists a linear continuous mappingDf (ϑ0) L2(FRd) rarrR (Df (ϑ0) isin L(L2(FRd)R)) such that
f (ϑ0 + η) minus f (ϑ0) = Df (ϑ0)(η) + o(|η|L2
)(22)
with |η|L2 rarr 0 for η isin L2(FRd) Since Df (ϑ0) isin L(L2(FRd)R) Rieszrsquo rep-resentation theorem yields the existence of a (P -as) unique random variable θ0 isinL2(FRd) such that Df (ϑ0)(η) = (θ0 η)L2 = E[θ0η] for all η isin L2(FRd) In[17] (see also [6]) it has been proved that there is a Borel function h0 Rd rarr R
d
such that θ0 = h0(ϑ0)P -as The function h0 only depends on the law Pϑ0 but noton ϑ0 itself Taking into account the definition of f this allows to write
f (Pϑ) minus f (Pϑ0) = E[h0(ϑ0) middot (ϑ minus ϑ0)
] + o(|ϑ minus ϑ0|L2
)(23)
ϑ isin L2(FRd)We call partμf (Pϑ0 y) = h0(y) y isin R
d the derivative of f P2(Rd) rarr R at
Pϑ0 Note that partμf (Pϑ0 y) is only Pϑ0(dy)-ae uniquely determinedHowever in our approach we have to consider functions f P2(R
d) rarrR whichare differentiable in all elements of P2(R
d) In order to simplify the argumentwe suppose that f L2(FRd) rarr R is Freacutechet differential over the whole spaceL2(FRd) This corresponds to a large class of important examples In this casewe have the derivative partμf (Pϑ y) defined Pϑ(dy)-ae for all ϑ isin L2(FRd)In Lemma 32 [9] it is shown that if furthermore the Freacutechet derivative Df L2(FRd) rarr L(L2(FRd)R) is Lipschitz continuous (with a Lipschitz constant
830 BUCKDAHN LI PENG AND RAINER
K isin R+) then there is for all ϑ isin L2(FRd) a Pϑ -version of partμf (Pϑ middot) Rd rarrR
d such that∣∣partμf (Pϑ y) minus partμf(Pϑ yprime)∣∣ le K
∣∣y minus yprime∣∣ for all y yprime isin Rd
This motivates us to make the following definition
DEFINITION 21 We say that f isin C11b (P2(R
d)) [continuously differentiableover P2(R
d) with Lipschitz-continuous bounded derivative] if there exists for allϑ isin L2(FRd) a Pϑ -modification of partμf (Pϑ middot) again denoted by partμf (Pϑ middot)such that partμf P2(R
d) timesRd rarr R
d is bounded and Lipschitz continuous that isthere is some real constant C such that
(i) |partμf (μx)| le Cμ isin P2(Rd) x isin R
d (ii) |partμf (μx) minus partμf (μprime xprime)| le C(W2(μμprime) + |x minus xprime|)μμprime isin P2(R
d) xxprime isin R
d
we consider this function partμf as the derivative of f
REMARK 21 Let us point out that if f isin C11b (P2(R
d)) the version ofpartμf (Pϑ middot) ϑ isin L2(FRd) indicated in Definition 21 is unique Indeed givenϑ isin L2(FRd) let θ be a d-dimensional vector of independent standard nor-mally distributed random variables which are independent of ϑ Then sincepartμf (Pϑ+εθ ϑ + εθ) is only P -as defined partμf (Pϑ+εθ y) is determined onlyPϑ+εθ (dy)-as Observing that the random variable ϑ +εθ possesses a strictly pos-itive density on R
d it follows that Pϑ+εθ and the Lebesgue measure over Rd areequivalent Consequently partμf (Pϑ+εθ y) is defined dy-ae on R
d From the Lip-schitz continuity (ii) of partμf in Definition 21 it then follows that partμf (Pϑ+εθ y)
is defined for all y isin Rd and taking the limit 0 lt ε darr 0 yields that partμf (Pϑ y) is
uniquely defined for all y isin Rd
Given now f isin C11b (P2(R
d)) for fixed y isin Rd the question of the differen-
tiability of its components (partμf )j (middot y) P2(Rd) rarr R1 le j le d raises and
it can be discussed in the same way as the first-order derivative partμf above If(partμf )j (middot y) P2(R
d) rarr R belongs to C11b (P2(R
d)) we have that its derivativepartμ((partμf )j (middot y))(middot middot) P2(R
d)timesRd rarrR
d is a Lipschitz-continuous function forevery y isin R
d Then
part2μf (μx y) = (
partμ
((partμf )j (middot y)
)(μ x)
)1lejled
(24)(μ x y) isin P2
(R
d) timesR
d timesRd
defines a function part2μf P2(R
d) timesRd timesR
d rarrRd otimesR
d
DEFINITION 22 We say that f isin C21b (P2(R
d)) if f isin C11b (P2(R
d)) and
MEAN-FIELD SDES AND ASSOCIATED PDES 831
(i) (partμf )j (middot y) isin C11b (P2(R
d)) for all y isin Rd1 le j le d and part2
μf P2(R
d) timesRd timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous(ii) partμf (μ middot) Rd rarr R
d is differentiable for every μ isin P2(Rd) and its
derivative partypartμf P2(Rd)timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous
Adopting the above introduced notation we consider a functionf isin C
21b (P2(R
d)) and discuss its second-order Taylor expansion For this endwe have still to introduce some notation Let ( F P ) be a copy of the prob-ability space (FP ) For any random variable (of arbitrary dimension) ϑ
over (FP ) we denote by ϑ a copy (of the same law as ϑ but definedover ( F P ) Pϑ = Pϑ The expectation E[middot] = int
(middot) dP acts only over thevariables endowed with a tilde This can be made rigorous by working withthe product space (FP ) otimes ( F P ) = (FP ) otimes (FP ) and puttingϑ(ωω) = ϑ(ω) (ωω) isin times = times for ϑ random variable defined over(FP )) Of course this formalism can be easily extended from random vari-ables to stochastic processes
With the above notation and writing a otimes b = (aibj )1leijled for a = (ai)1leiled
b = (bj )1lejled isin Rd we can state now the following result
LEMMA 21 Let f isin C21b (P2(R
d)) Then for any given ϑ0 isin L2(FRd) wehave the following second-order expansion
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](25)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R(PϑPϑ0)
ϑ isin L2(FRd
)
where η = ϑ minus ϑ0 and for all ϑ isin L2(FRd) the remainder R(PϑPϑ0) satisfiesthe estimate ∣∣R(PϑPϑ0)
∣∣ le C((
E[|ϑ minus ϑ0|2])32 + E
[|ϑ minus ϑ0|3])(26)
le CE[|ϑ minus ϑ0|3]
The constant C isin R+ only depends on the Lipschitz constants of part2μf and partypartμf
We observe that the above second-order expansion does not constitute a second-order Taylor expansion for the associated function f L2(FRd) rarr R since theremainder term E[|ϑ minus ϑ0|3 and |ϑ minus ϑ0|2] is not of order o(|ϑ minus ϑ0|2L2) Indeed asthe following example shows in general we only have E[|ϑ minusϑ0|3 and |ϑ minusϑ0|2] =O(|ϑ minus ϑ0|2L2)
832 BUCKDAHN LI PENG AND RAINER
EXAMPLE 21 Let ϑ = IA with A isin F such that P(A) rarr 0 as rarr infin
Then ϑ rarr 0 in L2( rarr infin) and E[|ϑ|3 and |ϑ|2] = P(A) = E[ϑ2 ] = |ϑ|2L2 rarr
0 ( rarr infin)
If we had in Lemma 21 a remainder R(PϑPϑ0) = o(|ϑ minus ϑ0|2L2) this wouldsuggest that f (ζ ) = f (Pζ ) is twice Freacutechet differentiable at ϑ0 But however asthe following Example 23 shows even in easiest cases f is not twice Freacutechetdifferentiable
However for our purposes the above expansion is fine
PROOF OF LEMMA 21 Let ϑ0 isin L2(FRd) Then for all ϑ isin L2(FRd)putting η = ϑ minus ϑ0 and using the fact that f isin C
21b (P2(R
d)) we have
f (Pϑ) minus f (Pϑ0) =int 1
0
d
dλf (Pϑ0+λη) dλ
(27)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
Let us now compute ddλ
partμf (Pϑ0+ληϑ0 +λη) Since f isin C21b (P2(R
d)) the liftedfunction partμf (ξ y) = partμf (Pξ y) ξ isin L2(FRd) is Freacutechet differentiable in ξ and
d
dλpartμf (Pϑ0+λη y) = d
dλpartμf (ϑ0 + λη y)
(28)= E
[part2μf (Pϑ0+ληϑ0 + λη y) middot η]
λ isin R y isin Rd Then choosing an independent copy (ϑ ϑ0) of (ϑϑ0) defined
over ( F P ) (ie in particular P(ϑϑ0)= P(ϑϑ0)) we have for η = ϑ minus ϑ0
d
dλpartμf (Pϑ0+λη y) = E
[part2μf (Pϑ0+λη ϑ0 + λη y) middot η]
(29)λ isin R y isin R
d
Consequently
d
dλpartμf (Pϑ0+ληϑ0 + λη) = E
[part2μf (Pϑ0+λη ϑ0 + ληϑ0 + λη) middot η]
(210)+ partypartμf (Pϑ0+ληϑ0 + λη) middot η
Thus
f (Pϑ) minus f (Pϑ0)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
MEAN-FIELD SDES AND ASSOCIATED PDES 833
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[d
dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η
]dρ dλ(211)
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[tr
(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η
)]dρ dλ
+int 1
0
int λ
0E
[E
[tr
(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η
)]]dρ dλ
From this latter relation we get
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](212)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R1(PϑPϑ0) + R2(PϑPϑ0)
with the remainders
R1(PϑPϑ0) =int 1
0
int λ
0E
[E
[tr
((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)
(213)minus part2
μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η
)]]dρ dλ
and
R2(PϑPϑ0) =int 1
0
int λ
0E
[tr
((partypartμf (Pϑ0+ρηϑ0 + ρη)
(214)minus partypartμf (Pϑ0 ϑ0)
) middot η otimes η)]
dρ dλ
Finally from the boundedness and the Lipschitz continuity of the functions part2μf
and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf
|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement
Let us finish our preliminary discussion with two illustrating examples
EXAMPLE 22 Given two twice continuously differentiable functions h R
d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and
f (ϑ0 + η) minus f (ϑ0) =int 1
0gprime(E[
h(ϑ0 + sη)])
E[hprime(ϑ0 + sη)η
]ds
= gprime(E[h(ϑ0)
])E
[hprime(ϑ0)η
] + o(|η|L2
)= E
[gprime(E[
h(ϑ0)])
hprime(ϑ0)η] + o
(|η|L2)
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
MEAN-FIELD SDES AND ASSOCIATED PDES 829
P2(R2d) is endowed with the 2-Wasserstein metric
W2(μ ν) = inf(int
RdtimesRd|x minus y|2ρ(dx dy)
)12
ρ isin P2(R
2d)(21)
with ρ(middot timesR
d) = μρ(R
d times middot) = ν
μ ν isin P2
(R
d)
Among the different notions of differentiability of a function f defined overP2(R
d) we adopt for our approach that introduced by Lions [17] in his lectures atCollegravege de France in Paris and revised in the notes by Cardaliaguet [6] we referthe reader also for instance to Carmona and Delarue [9] Let us consider a proba-bility space (FP ) which is ldquorich enoughrdquo (the precise space we will work withwill be introduced later) ldquoRich enoughrdquo means that for every μ isin P2(R
d) thereis a random variable ϑ isin L2(FRd)(= L2(FP Rd)) such that Pϑ = μ It iswell known that the probability space ([01]B([01]) dx) has this property
Identifying the random variables in L2(FRd) which coincide P -as we canregard L2(FRd) as a Hilbert space with inner product (ξ η)L2 = E[ξ middot η] ξ η isinL2(FRd) and norm |ξ |L2 = (ξ ξ)
12L2 Recall that due to the definition made
by Lions [6] (see Cardaliaguet [7]) a function f P2(Rd) rarr R is said to be dif-
ferentiable in μ isin P2(Rd) if for f (ϑ) = f (Pϑ)ϑ isin L2(FRd) there is some
ϑ0 isin L2(FRd) with Pϑ0 = μ such that the function f L2(FRd) rarr R is dif-ferentiable (in Freacutechet sense) in ϑ0 that is there exists a linear continuous mappingDf (ϑ0) L2(FRd) rarrR (Df (ϑ0) isin L(L2(FRd)R)) such that
f (ϑ0 + η) minus f (ϑ0) = Df (ϑ0)(η) + o(|η|L2
)(22)
with |η|L2 rarr 0 for η isin L2(FRd) Since Df (ϑ0) isin L(L2(FRd)R) Rieszrsquo rep-resentation theorem yields the existence of a (P -as) unique random variable θ0 isinL2(FRd) such that Df (ϑ0)(η) = (θ0 η)L2 = E[θ0η] for all η isin L2(FRd) In[17] (see also [6]) it has been proved that there is a Borel function h0 Rd rarr R
d
such that θ0 = h0(ϑ0)P -as The function h0 only depends on the law Pϑ0 but noton ϑ0 itself Taking into account the definition of f this allows to write
f (Pϑ) minus f (Pϑ0) = E[h0(ϑ0) middot (ϑ minus ϑ0)
] + o(|ϑ minus ϑ0|L2
)(23)
ϑ isin L2(FRd)We call partμf (Pϑ0 y) = h0(y) y isin R
d the derivative of f P2(Rd) rarr R at
Pϑ0 Note that partμf (Pϑ0 y) is only Pϑ0(dy)-ae uniquely determinedHowever in our approach we have to consider functions f P2(R
d) rarrR whichare differentiable in all elements of P2(R
d) In order to simplify the argumentwe suppose that f L2(FRd) rarr R is Freacutechet differential over the whole spaceL2(FRd) This corresponds to a large class of important examples In this casewe have the derivative partμf (Pϑ y) defined Pϑ(dy)-ae for all ϑ isin L2(FRd)In Lemma 32 [9] it is shown that if furthermore the Freacutechet derivative Df L2(FRd) rarr L(L2(FRd)R) is Lipschitz continuous (with a Lipschitz constant
830 BUCKDAHN LI PENG AND RAINER
K isin R+) then there is for all ϑ isin L2(FRd) a Pϑ -version of partμf (Pϑ middot) Rd rarrR
d such that∣∣partμf (Pϑ y) minus partμf(Pϑ yprime)∣∣ le K
∣∣y minus yprime∣∣ for all y yprime isin Rd
This motivates us to make the following definition
DEFINITION 21 We say that f isin C11b (P2(R
d)) [continuously differentiableover P2(R
d) with Lipschitz-continuous bounded derivative] if there exists for allϑ isin L2(FRd) a Pϑ -modification of partμf (Pϑ middot) again denoted by partμf (Pϑ middot)such that partμf P2(R
d) timesRd rarr R
d is bounded and Lipschitz continuous that isthere is some real constant C such that
(i) |partμf (μx)| le Cμ isin P2(Rd) x isin R
d (ii) |partμf (μx) minus partμf (μprime xprime)| le C(W2(μμprime) + |x minus xprime|)μμprime isin P2(R
d) xxprime isin R
d
we consider this function partμf as the derivative of f
REMARK 21 Let us point out that if f isin C11b (P2(R
d)) the version ofpartμf (Pϑ middot) ϑ isin L2(FRd) indicated in Definition 21 is unique Indeed givenϑ isin L2(FRd) let θ be a d-dimensional vector of independent standard nor-mally distributed random variables which are independent of ϑ Then sincepartμf (Pϑ+εθ ϑ + εθ) is only P -as defined partμf (Pϑ+εθ y) is determined onlyPϑ+εθ (dy)-as Observing that the random variable ϑ +εθ possesses a strictly pos-itive density on R
d it follows that Pϑ+εθ and the Lebesgue measure over Rd areequivalent Consequently partμf (Pϑ+εθ y) is defined dy-ae on R
d From the Lip-schitz continuity (ii) of partμf in Definition 21 it then follows that partμf (Pϑ+εθ y)
is defined for all y isin Rd and taking the limit 0 lt ε darr 0 yields that partμf (Pϑ y) is
uniquely defined for all y isin Rd
Given now f isin C11b (P2(R
d)) for fixed y isin Rd the question of the differen-
tiability of its components (partμf )j (middot y) P2(Rd) rarr R1 le j le d raises and
it can be discussed in the same way as the first-order derivative partμf above If(partμf )j (middot y) P2(R
d) rarr R belongs to C11b (P2(R
d)) we have that its derivativepartμ((partμf )j (middot y))(middot middot) P2(R
d)timesRd rarrR
d is a Lipschitz-continuous function forevery y isin R
d Then
part2μf (μx y) = (
partμ
((partμf )j (middot y)
)(μ x)
)1lejled
(24)(μ x y) isin P2
(R
d) timesR
d timesRd
defines a function part2μf P2(R
d) timesRd timesR
d rarrRd otimesR
d
DEFINITION 22 We say that f isin C21b (P2(R
d)) if f isin C11b (P2(R
d)) and
MEAN-FIELD SDES AND ASSOCIATED PDES 831
(i) (partμf )j (middot y) isin C11b (P2(R
d)) for all y isin Rd1 le j le d and part2
μf P2(R
d) timesRd timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous(ii) partμf (μ middot) Rd rarr R
d is differentiable for every μ isin P2(Rd) and its
derivative partypartμf P2(Rd)timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous
Adopting the above introduced notation we consider a functionf isin C
21b (P2(R
d)) and discuss its second-order Taylor expansion For this endwe have still to introduce some notation Let ( F P ) be a copy of the prob-ability space (FP ) For any random variable (of arbitrary dimension) ϑ
over (FP ) we denote by ϑ a copy (of the same law as ϑ but definedover ( F P ) Pϑ = Pϑ The expectation E[middot] = int
(middot) dP acts only over thevariables endowed with a tilde This can be made rigorous by working withthe product space (FP ) otimes ( F P ) = (FP ) otimes (FP ) and puttingϑ(ωω) = ϑ(ω) (ωω) isin times = times for ϑ random variable defined over(FP )) Of course this formalism can be easily extended from random vari-ables to stochastic processes
With the above notation and writing a otimes b = (aibj )1leijled for a = (ai)1leiled
b = (bj )1lejled isin Rd we can state now the following result
LEMMA 21 Let f isin C21b (P2(R
d)) Then for any given ϑ0 isin L2(FRd) wehave the following second-order expansion
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](25)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R(PϑPϑ0)
ϑ isin L2(FRd
)
where η = ϑ minus ϑ0 and for all ϑ isin L2(FRd) the remainder R(PϑPϑ0) satisfiesthe estimate ∣∣R(PϑPϑ0)
∣∣ le C((
E[|ϑ minus ϑ0|2])32 + E
[|ϑ minus ϑ0|3])(26)
le CE[|ϑ minus ϑ0|3]
The constant C isin R+ only depends on the Lipschitz constants of part2μf and partypartμf
We observe that the above second-order expansion does not constitute a second-order Taylor expansion for the associated function f L2(FRd) rarr R since theremainder term E[|ϑ minus ϑ0|3 and |ϑ minus ϑ0|2] is not of order o(|ϑ minus ϑ0|2L2) Indeed asthe following example shows in general we only have E[|ϑ minusϑ0|3 and |ϑ minusϑ0|2] =O(|ϑ minus ϑ0|2L2)
832 BUCKDAHN LI PENG AND RAINER
EXAMPLE 21 Let ϑ = IA with A isin F such that P(A) rarr 0 as rarr infin
Then ϑ rarr 0 in L2( rarr infin) and E[|ϑ|3 and |ϑ|2] = P(A) = E[ϑ2 ] = |ϑ|2L2 rarr
0 ( rarr infin)
If we had in Lemma 21 a remainder R(PϑPϑ0) = o(|ϑ minus ϑ0|2L2) this wouldsuggest that f (ζ ) = f (Pζ ) is twice Freacutechet differentiable at ϑ0 But however asthe following Example 23 shows even in easiest cases f is not twice Freacutechetdifferentiable
However for our purposes the above expansion is fine
PROOF OF LEMMA 21 Let ϑ0 isin L2(FRd) Then for all ϑ isin L2(FRd)putting η = ϑ minus ϑ0 and using the fact that f isin C
21b (P2(R
d)) we have
f (Pϑ) minus f (Pϑ0) =int 1
0
d
dλf (Pϑ0+λη) dλ
(27)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
Let us now compute ddλ
partμf (Pϑ0+ληϑ0 +λη) Since f isin C21b (P2(R
d)) the liftedfunction partμf (ξ y) = partμf (Pξ y) ξ isin L2(FRd) is Freacutechet differentiable in ξ and
d
dλpartμf (Pϑ0+λη y) = d
dλpartμf (ϑ0 + λη y)
(28)= E
[part2μf (Pϑ0+ληϑ0 + λη y) middot η]
λ isin R y isin Rd Then choosing an independent copy (ϑ ϑ0) of (ϑϑ0) defined
over ( F P ) (ie in particular P(ϑϑ0)= P(ϑϑ0)) we have for η = ϑ minus ϑ0
d
dλpartμf (Pϑ0+λη y) = E
[part2μf (Pϑ0+λη ϑ0 + λη y) middot η]
(29)λ isin R y isin R
d
Consequently
d
dλpartμf (Pϑ0+ληϑ0 + λη) = E
[part2μf (Pϑ0+λη ϑ0 + ληϑ0 + λη) middot η]
(210)+ partypartμf (Pϑ0+ληϑ0 + λη) middot η
Thus
f (Pϑ) minus f (Pϑ0)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
MEAN-FIELD SDES AND ASSOCIATED PDES 833
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[d
dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η
]dρ dλ(211)
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[tr
(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η
)]dρ dλ
+int 1
0
int λ
0E
[E
[tr
(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η
)]]dρ dλ
From this latter relation we get
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](212)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R1(PϑPϑ0) + R2(PϑPϑ0)
with the remainders
R1(PϑPϑ0) =int 1
0
int λ
0E
[E
[tr
((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)
(213)minus part2
μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η
)]]dρ dλ
and
R2(PϑPϑ0) =int 1
0
int λ
0E
[tr
((partypartμf (Pϑ0+ρηϑ0 + ρη)
(214)minus partypartμf (Pϑ0 ϑ0)
) middot η otimes η)]
dρ dλ
Finally from the boundedness and the Lipschitz continuity of the functions part2μf
and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf
|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement
Let us finish our preliminary discussion with two illustrating examples
EXAMPLE 22 Given two twice continuously differentiable functions h R
d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and
f (ϑ0 + η) minus f (ϑ0) =int 1
0gprime(E[
h(ϑ0 + sη)])
E[hprime(ϑ0 + sη)η
]ds
= gprime(E[h(ϑ0)
])E
[hprime(ϑ0)η
] + o(|η|L2
)= E
[gprime(E[
h(ϑ0)])
hprime(ϑ0)η] + o
(|η|L2)
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
830 BUCKDAHN LI PENG AND RAINER
K isin R+) then there is for all ϑ isin L2(FRd) a Pϑ -version of partμf (Pϑ middot) Rd rarrR
d such that∣∣partμf (Pϑ y) minus partμf(Pϑ yprime)∣∣ le K
∣∣y minus yprime∣∣ for all y yprime isin Rd
This motivates us to make the following definition
DEFINITION 21 We say that f isin C11b (P2(R
d)) [continuously differentiableover P2(R
d) with Lipschitz-continuous bounded derivative] if there exists for allϑ isin L2(FRd) a Pϑ -modification of partμf (Pϑ middot) again denoted by partμf (Pϑ middot)such that partμf P2(R
d) timesRd rarr R
d is bounded and Lipschitz continuous that isthere is some real constant C such that
(i) |partμf (μx)| le Cμ isin P2(Rd) x isin R
d (ii) |partμf (μx) minus partμf (μprime xprime)| le C(W2(μμprime) + |x minus xprime|)μμprime isin P2(R
d) xxprime isin R
d
we consider this function partμf as the derivative of f
REMARK 21 Let us point out that if f isin C11b (P2(R
d)) the version ofpartμf (Pϑ middot) ϑ isin L2(FRd) indicated in Definition 21 is unique Indeed givenϑ isin L2(FRd) let θ be a d-dimensional vector of independent standard nor-mally distributed random variables which are independent of ϑ Then sincepartμf (Pϑ+εθ ϑ + εθ) is only P -as defined partμf (Pϑ+εθ y) is determined onlyPϑ+εθ (dy)-as Observing that the random variable ϑ +εθ possesses a strictly pos-itive density on R
d it follows that Pϑ+εθ and the Lebesgue measure over Rd areequivalent Consequently partμf (Pϑ+εθ y) is defined dy-ae on R
d From the Lip-schitz continuity (ii) of partμf in Definition 21 it then follows that partμf (Pϑ+εθ y)
is defined for all y isin Rd and taking the limit 0 lt ε darr 0 yields that partμf (Pϑ y) is
uniquely defined for all y isin Rd
Given now f isin C11b (P2(R
d)) for fixed y isin Rd the question of the differen-
tiability of its components (partμf )j (middot y) P2(Rd) rarr R1 le j le d raises and
it can be discussed in the same way as the first-order derivative partμf above If(partμf )j (middot y) P2(R
d) rarr R belongs to C11b (P2(R
d)) we have that its derivativepartμ((partμf )j (middot y))(middot middot) P2(R
d)timesRd rarrR
d is a Lipschitz-continuous function forevery y isin R
d Then
part2μf (μx y) = (
partμ
((partμf )j (middot y)
)(μ x)
)1lejled
(24)(μ x y) isin P2
(R
d) timesR
d timesRd
defines a function part2μf P2(R
d) timesRd timesR
d rarrRd otimesR
d
DEFINITION 22 We say that f isin C21b (P2(R
d)) if f isin C11b (P2(R
d)) and
MEAN-FIELD SDES AND ASSOCIATED PDES 831
(i) (partμf )j (middot y) isin C11b (P2(R
d)) for all y isin Rd1 le j le d and part2
μf P2(R
d) timesRd timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous(ii) partμf (μ middot) Rd rarr R
d is differentiable for every μ isin P2(Rd) and its
derivative partypartμf P2(Rd)timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous
Adopting the above introduced notation we consider a functionf isin C
21b (P2(R
d)) and discuss its second-order Taylor expansion For this endwe have still to introduce some notation Let ( F P ) be a copy of the prob-ability space (FP ) For any random variable (of arbitrary dimension) ϑ
over (FP ) we denote by ϑ a copy (of the same law as ϑ but definedover ( F P ) Pϑ = Pϑ The expectation E[middot] = int
(middot) dP acts only over thevariables endowed with a tilde This can be made rigorous by working withthe product space (FP ) otimes ( F P ) = (FP ) otimes (FP ) and puttingϑ(ωω) = ϑ(ω) (ωω) isin times = times for ϑ random variable defined over(FP )) Of course this formalism can be easily extended from random vari-ables to stochastic processes
With the above notation and writing a otimes b = (aibj )1leijled for a = (ai)1leiled
b = (bj )1lejled isin Rd we can state now the following result
LEMMA 21 Let f isin C21b (P2(R
d)) Then for any given ϑ0 isin L2(FRd) wehave the following second-order expansion
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](25)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R(PϑPϑ0)
ϑ isin L2(FRd
)
where η = ϑ minus ϑ0 and for all ϑ isin L2(FRd) the remainder R(PϑPϑ0) satisfiesthe estimate ∣∣R(PϑPϑ0)
∣∣ le C((
E[|ϑ minus ϑ0|2])32 + E
[|ϑ minus ϑ0|3])(26)
le CE[|ϑ minus ϑ0|3]
The constant C isin R+ only depends on the Lipschitz constants of part2μf and partypartμf
We observe that the above second-order expansion does not constitute a second-order Taylor expansion for the associated function f L2(FRd) rarr R since theremainder term E[|ϑ minus ϑ0|3 and |ϑ minus ϑ0|2] is not of order o(|ϑ minus ϑ0|2L2) Indeed asthe following example shows in general we only have E[|ϑ minusϑ0|3 and |ϑ minusϑ0|2] =O(|ϑ minus ϑ0|2L2)
832 BUCKDAHN LI PENG AND RAINER
EXAMPLE 21 Let ϑ = IA with A isin F such that P(A) rarr 0 as rarr infin
Then ϑ rarr 0 in L2( rarr infin) and E[|ϑ|3 and |ϑ|2] = P(A) = E[ϑ2 ] = |ϑ|2L2 rarr
0 ( rarr infin)
If we had in Lemma 21 a remainder R(PϑPϑ0) = o(|ϑ minus ϑ0|2L2) this wouldsuggest that f (ζ ) = f (Pζ ) is twice Freacutechet differentiable at ϑ0 But however asthe following Example 23 shows even in easiest cases f is not twice Freacutechetdifferentiable
However for our purposes the above expansion is fine
PROOF OF LEMMA 21 Let ϑ0 isin L2(FRd) Then for all ϑ isin L2(FRd)putting η = ϑ minus ϑ0 and using the fact that f isin C
21b (P2(R
d)) we have
f (Pϑ) minus f (Pϑ0) =int 1
0
d
dλf (Pϑ0+λη) dλ
(27)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
Let us now compute ddλ
partμf (Pϑ0+ληϑ0 +λη) Since f isin C21b (P2(R
d)) the liftedfunction partμf (ξ y) = partμf (Pξ y) ξ isin L2(FRd) is Freacutechet differentiable in ξ and
d
dλpartμf (Pϑ0+λη y) = d
dλpartμf (ϑ0 + λη y)
(28)= E
[part2μf (Pϑ0+ληϑ0 + λη y) middot η]
λ isin R y isin Rd Then choosing an independent copy (ϑ ϑ0) of (ϑϑ0) defined
over ( F P ) (ie in particular P(ϑϑ0)= P(ϑϑ0)) we have for η = ϑ minus ϑ0
d
dλpartμf (Pϑ0+λη y) = E
[part2μf (Pϑ0+λη ϑ0 + λη y) middot η]
(29)λ isin R y isin R
d
Consequently
d
dλpartμf (Pϑ0+ληϑ0 + λη) = E
[part2μf (Pϑ0+λη ϑ0 + ληϑ0 + λη) middot η]
(210)+ partypartμf (Pϑ0+ληϑ0 + λη) middot η
Thus
f (Pϑ) minus f (Pϑ0)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
MEAN-FIELD SDES AND ASSOCIATED PDES 833
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[d
dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η
]dρ dλ(211)
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[tr
(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η
)]dρ dλ
+int 1
0
int λ
0E
[E
[tr
(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η
)]]dρ dλ
From this latter relation we get
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](212)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R1(PϑPϑ0) + R2(PϑPϑ0)
with the remainders
R1(PϑPϑ0) =int 1
0
int λ
0E
[E
[tr
((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)
(213)minus part2
μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η
)]]dρ dλ
and
R2(PϑPϑ0) =int 1
0
int λ
0E
[tr
((partypartμf (Pϑ0+ρηϑ0 + ρη)
(214)minus partypartμf (Pϑ0 ϑ0)
) middot η otimes η)]
dρ dλ
Finally from the boundedness and the Lipschitz continuity of the functions part2μf
and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf
|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement
Let us finish our preliminary discussion with two illustrating examples
EXAMPLE 22 Given two twice continuously differentiable functions h R
d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and
f (ϑ0 + η) minus f (ϑ0) =int 1
0gprime(E[
h(ϑ0 + sη)])
E[hprime(ϑ0 + sη)η
]ds
= gprime(E[h(ϑ0)
])E
[hprime(ϑ0)η
] + o(|η|L2
)= E
[gprime(E[
h(ϑ0)])
hprime(ϑ0)η] + o
(|η|L2)
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
MEAN-FIELD SDES AND ASSOCIATED PDES 831
(i) (partμf )j (middot y) isin C11b (P2(R
d)) for all y isin Rd1 le j le d and part2
μf P2(R
d) timesRd timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous(ii) partμf (μ middot) Rd rarr R
d is differentiable for every μ isin P2(Rd) and its
derivative partypartμf P2(Rd)timesR
d rarr Rd otimesR
d is bounded and Lipschitz-continuous
Adopting the above introduced notation we consider a functionf isin C
21b (P2(R
d)) and discuss its second-order Taylor expansion For this endwe have still to introduce some notation Let ( F P ) be a copy of the prob-ability space (FP ) For any random variable (of arbitrary dimension) ϑ
over (FP ) we denote by ϑ a copy (of the same law as ϑ but definedover ( F P ) Pϑ = Pϑ The expectation E[middot] = int
(middot) dP acts only over thevariables endowed with a tilde This can be made rigorous by working withthe product space (FP ) otimes ( F P ) = (FP ) otimes (FP ) and puttingϑ(ωω) = ϑ(ω) (ωω) isin times = times for ϑ random variable defined over(FP )) Of course this formalism can be easily extended from random vari-ables to stochastic processes
With the above notation and writing a otimes b = (aibj )1leijled for a = (ai)1leiled
b = (bj )1lejled isin Rd we can state now the following result
LEMMA 21 Let f isin C21b (P2(R
d)) Then for any given ϑ0 isin L2(FRd) wehave the following second-order expansion
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](25)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R(PϑPϑ0)
ϑ isin L2(FRd
)
where η = ϑ minus ϑ0 and for all ϑ isin L2(FRd) the remainder R(PϑPϑ0) satisfiesthe estimate ∣∣R(PϑPϑ0)
∣∣ le C((
E[|ϑ minus ϑ0|2])32 + E
[|ϑ minus ϑ0|3])(26)
le CE[|ϑ minus ϑ0|3]
The constant C isin R+ only depends on the Lipschitz constants of part2μf and partypartμf
We observe that the above second-order expansion does not constitute a second-order Taylor expansion for the associated function f L2(FRd) rarr R since theremainder term E[|ϑ minus ϑ0|3 and |ϑ minus ϑ0|2] is not of order o(|ϑ minus ϑ0|2L2) Indeed asthe following example shows in general we only have E[|ϑ minusϑ0|3 and |ϑ minusϑ0|2] =O(|ϑ minus ϑ0|2L2)
832 BUCKDAHN LI PENG AND RAINER
EXAMPLE 21 Let ϑ = IA with A isin F such that P(A) rarr 0 as rarr infin
Then ϑ rarr 0 in L2( rarr infin) and E[|ϑ|3 and |ϑ|2] = P(A) = E[ϑ2 ] = |ϑ|2L2 rarr
0 ( rarr infin)
If we had in Lemma 21 a remainder R(PϑPϑ0) = o(|ϑ minus ϑ0|2L2) this wouldsuggest that f (ζ ) = f (Pζ ) is twice Freacutechet differentiable at ϑ0 But however asthe following Example 23 shows even in easiest cases f is not twice Freacutechetdifferentiable
However for our purposes the above expansion is fine
PROOF OF LEMMA 21 Let ϑ0 isin L2(FRd) Then for all ϑ isin L2(FRd)putting η = ϑ minus ϑ0 and using the fact that f isin C
21b (P2(R
d)) we have
f (Pϑ) minus f (Pϑ0) =int 1
0
d
dλf (Pϑ0+λη) dλ
(27)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
Let us now compute ddλ
partμf (Pϑ0+ληϑ0 +λη) Since f isin C21b (P2(R
d)) the liftedfunction partμf (ξ y) = partμf (Pξ y) ξ isin L2(FRd) is Freacutechet differentiable in ξ and
d
dλpartμf (Pϑ0+λη y) = d
dλpartμf (ϑ0 + λη y)
(28)= E
[part2μf (Pϑ0+ληϑ0 + λη y) middot η]
λ isin R y isin Rd Then choosing an independent copy (ϑ ϑ0) of (ϑϑ0) defined
over ( F P ) (ie in particular P(ϑϑ0)= P(ϑϑ0)) we have for η = ϑ minus ϑ0
d
dλpartμf (Pϑ0+λη y) = E
[part2μf (Pϑ0+λη ϑ0 + λη y) middot η]
(29)λ isin R y isin R
d
Consequently
d
dλpartμf (Pϑ0+ληϑ0 + λη) = E
[part2μf (Pϑ0+λη ϑ0 + ληϑ0 + λη) middot η]
(210)+ partypartμf (Pϑ0+ληϑ0 + λη) middot η
Thus
f (Pϑ) minus f (Pϑ0)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
MEAN-FIELD SDES AND ASSOCIATED PDES 833
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[d
dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η
]dρ dλ(211)
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[tr
(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η
)]dρ dλ
+int 1
0
int λ
0E
[E
[tr
(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η
)]]dρ dλ
From this latter relation we get
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](212)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R1(PϑPϑ0) + R2(PϑPϑ0)
with the remainders
R1(PϑPϑ0) =int 1
0
int λ
0E
[E
[tr
((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)
(213)minus part2
μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η
)]]dρ dλ
and
R2(PϑPϑ0) =int 1
0
int λ
0E
[tr
((partypartμf (Pϑ0+ρηϑ0 + ρη)
(214)minus partypartμf (Pϑ0 ϑ0)
) middot η otimes η)]
dρ dλ
Finally from the boundedness and the Lipschitz continuity of the functions part2μf
and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf
|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement
Let us finish our preliminary discussion with two illustrating examples
EXAMPLE 22 Given two twice continuously differentiable functions h R
d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and
f (ϑ0 + η) minus f (ϑ0) =int 1
0gprime(E[
h(ϑ0 + sη)])
E[hprime(ϑ0 + sη)η
]ds
= gprime(E[h(ϑ0)
])E
[hprime(ϑ0)η
] + o(|η|L2
)= E
[gprime(E[
h(ϑ0)])
hprime(ϑ0)η] + o
(|η|L2)
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
832 BUCKDAHN LI PENG AND RAINER
EXAMPLE 21 Let ϑ = IA with A isin F such that P(A) rarr 0 as rarr infin
Then ϑ rarr 0 in L2( rarr infin) and E[|ϑ|3 and |ϑ|2] = P(A) = E[ϑ2 ] = |ϑ|2L2 rarr
0 ( rarr infin)
If we had in Lemma 21 a remainder R(PϑPϑ0) = o(|ϑ minus ϑ0|2L2) this wouldsuggest that f (ζ ) = f (Pζ ) is twice Freacutechet differentiable at ϑ0 But however asthe following Example 23 shows even in easiest cases f is not twice Freacutechetdifferentiable
However for our purposes the above expansion is fine
PROOF OF LEMMA 21 Let ϑ0 isin L2(FRd) Then for all ϑ isin L2(FRd)putting η = ϑ minus ϑ0 and using the fact that f isin C
21b (P2(R
d)) we have
f (Pϑ) minus f (Pϑ0) =int 1
0
d
dλf (Pϑ0+λη) dλ
(27)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
Let us now compute ddλ
partμf (Pϑ0+ληϑ0 +λη) Since f isin C21b (P2(R
d)) the liftedfunction partμf (ξ y) = partμf (Pξ y) ξ isin L2(FRd) is Freacutechet differentiable in ξ and
d
dλpartμf (Pϑ0+λη y) = d
dλpartμf (ϑ0 + λη y)
(28)= E
[part2μf (Pϑ0+ληϑ0 + λη y) middot η]
λ isin R y isin Rd Then choosing an independent copy (ϑ ϑ0) of (ϑϑ0) defined
over ( F P ) (ie in particular P(ϑϑ0)= P(ϑϑ0)) we have for η = ϑ minus ϑ0
d
dλpartμf (Pϑ0+λη y) = E
[part2μf (Pϑ0+λη ϑ0 + λη y) middot η]
(29)λ isin R y isin R
d
Consequently
d
dλpartμf (Pϑ0+ληϑ0 + λη) = E
[part2μf (Pϑ0+λη ϑ0 + ληϑ0 + λη) middot η]
(210)+ partypartμf (Pϑ0+ληϑ0 + λη) middot η
Thus
f (Pϑ) minus f (Pϑ0)
=int 1
0E
[partμf (Pϑ0+ληϑ0 + λη) middot η]
dλ
MEAN-FIELD SDES AND ASSOCIATED PDES 833
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[d
dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η
]dρ dλ(211)
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[tr
(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η
)]dρ dλ
+int 1
0
int λ
0E
[E
[tr
(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η
)]]dρ dλ
From this latter relation we get
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](212)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R1(PϑPϑ0) + R2(PϑPϑ0)
with the remainders
R1(PϑPϑ0) =int 1
0
int λ
0E
[E
[tr
((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)
(213)minus part2
μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η
)]]dρ dλ
and
R2(PϑPϑ0) =int 1
0
int λ
0E
[tr
((partypartμf (Pϑ0+ρηϑ0 + ρη)
(214)minus partypartμf (Pϑ0 ϑ0)
) middot η otimes η)]
dρ dλ
Finally from the boundedness and the Lipschitz continuity of the functions part2μf
and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf
|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement
Let us finish our preliminary discussion with two illustrating examples
EXAMPLE 22 Given two twice continuously differentiable functions h R
d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and
f (ϑ0 + η) minus f (ϑ0) =int 1
0gprime(E[
h(ϑ0 + sη)])
E[hprime(ϑ0 + sη)η
]ds
= gprime(E[h(ϑ0)
])E
[hprime(ϑ0)η
] + o(|η|L2
)= E
[gprime(E[
h(ϑ0)])
hprime(ϑ0)η] + o
(|η|L2)
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
MEAN-FIELD SDES AND ASSOCIATED PDES 833
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[d
dρpartμf (Pϑ0+ρηϑ0 + ρη) middot η
]dρ dλ(211)
= E[partμf (Pϑ0 ϑ0) middot η]
+int 1
0
int λ
0E
[tr
(partypartμf (Pϑ0+ρηϑ0 + ρη) middot η otimes η
)]dρ dλ
+int 1
0
int λ
0E
[E
[tr
(part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη) middot η otimes η
)]]dρ dλ
From this latter relation we get
f (Pϑ) minus f (Pϑ0)
= E[partμf (Pϑ0 ϑ0) middot η] + 1
2E[E
[tr
(part2μf (Pϑ0 ϑ0 ϑ0) middot η otimes η
)]](212)
+ 12E
[tr
(partypartμf (Pϑ0 ϑ0) middot η otimes η
)] + R1(PϑPϑ0) + R2(PϑPϑ0)
with the remainders
R1(PϑPϑ0) =int 1
0
int λ
0E
[E
[tr
((part2μf (Pϑ0+ρη ϑ0 + ρηϑ0 + ρη)
(213)minus part2
μf (Pϑ0 ϑ0 ϑ0)) middot η otimes η
)]]dρ dλ
and
R2(PϑPϑ0) =int 1
0
int λ
0E
[tr
((partypartμf (Pϑ0+ρηϑ0 + ρη)
(214)minus partypartμf (Pϑ0 ϑ0)
) middot η otimes η)]
dρ dλ
Finally from the boundedness and the Lipschitz continuity of the functions part2μf
and partypartμf we conclude that for some C isin R+ only depending on part2μf partypartμf
|R1(PϑPϑ0)| le C(E[|η|2])32 while |R2(PϑPϑ0)| le C(E|η|2)32 + CE|η|3This proves the statement
Let us finish our preliminary discussion with two illustrating examples
EXAMPLE 22 Given two twice continuously differentiable functions h R
d rarr R and g R rarr R with bounded derivatives we consider f (Pϑ) =g(E[h(ϑ)]) ϑ isin L2(FRd) Then given any ϑ0 isin L2(FRd) f (ϑ) = f (Pϑ) =g(E[h(ϑ)]) is Freacutechet differentiable in ϑ0 and
f (ϑ0 + η) minus f (ϑ0) =int 1
0gprime(E[
h(ϑ0 + sη)])
E[hprime(ϑ0 + sη)η
]ds
= gprime(E[h(ϑ0)
])E
[hprime(ϑ0)η
] + o(|η|L2
)= E
[gprime(E[
h(ϑ0)])
hprime(ϑ0)η] + o
(|η|L2)
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
834 BUCKDAHN LI PENG AND RAINER
Thus Df (ϑ0)(η) = E[gprime(E[h(ϑ0)])partyh(ϑ0)η] η isin L2(FRd) that is
partμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(partyh)(y) y isin R
d
With the same argument we see that
part2μf (Pϑ0 x y) = gprimeprime(E[
h(ϑ0)])
(partxh)(x) otimes (partyh)(y)
and
partypartμf (Pϑ0 y) = gprime(E[h(ϑ0)
])(part2yh
)(y)
Consequently if g and h are three times continuously differentiable with boundedderivatives of all order then the second-order expansion stated in the aboveLemma 21 takes for this example the special form
g(E
[h(ϑ)
]) minus g(E
[h(ϑ0)
])= gprime(E[
h(ϑ0)])
E[partyh(ϑ0) middot (ϑ minus ϑ0)
]+ 1
2gprimeprime(E[h(ϑ0)
])(E
[partyh(ϑ0) middot (ϑ minus ϑ0)
])2
+ 12gprime(E[
h(ϑ0)])
E[tr
(part2yh(ϑ0) middot (ϑ minus ϑ0) otimes (ϑ minus ϑ0)
)]+ O
((E
[|ϑ minus ϑ0|2])32 + E[|ϑ minus ϑ0|3])
EXAMPLE 23 Let us consider a special case of Example 22 For d = 1 wechoose g(x) = x x isin R and h(x) = eix x isin R Then f (ϑ) = f (Pϑ) = ϕϑ(1) =E[eiϑ ] is just the characteristic function of ϑ at 1 Let ϑ0 isin L2(FR) be arbitraryand A isinF independent of ϑ0 We put η = IA and ϑ = ϑ0 +η Obviously the first-order Freacutechet derivative of f at ϑ0 is Dϑf (ϑ0)(η) = iE[eiϑ0η] and if f weretwice Freacutechet differentiable we would have
D2ϑ f (ϑ0)(η η) = Dϑ
[Dϑf (ϑ)
](η)|ϑ=ϑ0 = minusE
[eiϑ0
]η2
Then
R(PϑPϑ0) = f (ϑ) minus [f (ϑ0) + Dϑf (ϑ0)(η) + 1
2D2ϑf (ϑ0)(η η)
]= E
[eiϑ0
(eiη minus [
1 + iη minus 12η2])] = E
[eiϑ0
(ei minus (1
2 + i))
IA
]= (
ei minus (12 + i
))ϕϑ0(1)P (A) = (
ei minus (12 + i
))ϕϑ0(1)|η|2
L2
as |η|L2 = P(A)12 rarr 0 But this means that f is not twice Freacutechet differentiablein ϑ0 and taking into account the arbitrariness of ϑ0 isin L2(FR) we see that f isnowhere twice Freacutechet differentiable
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
MEAN-FIELD SDES AND ASSOCIATED PDES 835
3 The mean-field stochastic differential equation Let us now consider acomplete probability space (FP ) on which is defined a d-dimensional Brow-nian motion B(= (B1 Bd)) = (Bt )tisin[0T ] and T gt 0 denotes an arbitrarilyfixed time horizon We suppose that there is a sub-σ -field F0 sub F such that
(i) the Brownian motion B is independent of F0 and(ii) F0 is ldquorich enoughrdquo that is P2(R
k) = Pϑϑ isin L2(F0Rk) k ge 1
By F = (Ft )tisin[0T ] we denote the filtration generated by B completed and aug-mented by F0
Given deterministic Lipschitz functions σ Rd times P2(Rd) rarr R
dtimesd and b R
d times P2(Rd) rarr R
d we consider for the initial data (t x) isin [0 T ] times Rd and
ξ isin L2(Ft Rd) the stochastic differential equations (SDEs)
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
(31)s isin [t T ]
and
Xtxξs = x +
int s
tσ
(Xtxξ
r PX
tξr
)dBr +
int s
tb(Xtxξ
r PX
tξr
)dr
(32)s isin [t T ]
We observe that under our Lipschitz assumption on the coefficients the both SDEshave a unique solution in S2([t T ]Rd) the space of F-adapted continuous pro-cesses Y = (Ys)sisin[tT ] with the property E[supsisin[tT ] |Ys |2] lt +infin (see eg Car-mona and Delarue [9]) We see in particular that the solution Xtξ of the firstequation allows to determine that of the second equation As SDE standard esti-mates show we have for some C isin R+ depending only on the Lipschitz constantsof σ and b
E[
supsisin[tT ]
∣∣Xtxξs minus Xtxprimeξ
s
∣∣2]le C
∣∣x minus xprime∣∣2(33)
for all t isin [0 T ] x xprime isin Rd ξ isin L2(Ft Rd) This allows to substitute in the sec-
ond SDE for x the random variable ξ and shows that Xtxξ |x=ξ solves the sameSDE as Xtξ From the uniqueness of the solution we conclude
Xtxξs |x=ξ = Xtξ
s s isin [t T ](34)
Moreover from the uniqueness of the solution of the both equations we deduce thefollowing flow property(
XsXtxξs X
tξs
r XsXtξs
r
) = (Xtxξ
r Xtξr
) r isin [s T ](35)
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
836 BUCKDAHN LI PENG AND RAINER
for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) In fact putting η = X
tξs isin
L2(FsRd) and considering the SDEs (31) and (32) with the initial data (s y)
and (s η) respectively
Xsηr = η +
int r
sσ
(Xsη
u PXsηu
)dBu +
int r
sb(Xsη
u PXsηu
)du r isin [s T ]
and
Xsyηr = y +
int r
sσ
(Xsyη
u PXsηu
)dBu +
int r
sb(Xsyη
u PXsηu
)du r isin [s T ]
we get from the uniqueness of the solution that Xsηr = X
tξr r isin [s T ] and con-
sequently XsX
txξs η
r = Xtxξr r isin [t T ] that is we have (35)
Having this flow property it is natural to define for a sufficiently regular function Rd timesP2(R
d) rarrR an associated value function
V (t x ξ) = E[
(X
txξT P
XtξT
)]
(36)(t x) isin [0 T ] timesR
d ξ isin L2(Ft Rd
)
and to ask which PDE is satisfied by this function V In order to be able to answerthis question in the frame of the concept we have introduced above we have toshow that the function V (t x ξ) does not depend on ξ itself but only on its lawPξ that is that we have to do with a function V [0 T ] timesR
d timesP2(Rd) rarrR For
this the following lemma is useful
LEMMA 31 For all p ge 2 there is a constant Cp isin R+ only depending onthe Lipschitz constants of σ and b such that we have the following estimate
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(37)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd)
PROOF Recall that for the 2-Wasserstein metric W2(middot middot) we have
W2(PϑPθ ) = inf(
E[∣∣ϑ prime minus θ prime∣∣2])12
for all ϑ prime θ prime isin L2(F0Rd)
with Pϑ prime = PϑPθ prime = Pθ
(38)
le (E
[|ϑ minus θ |2])12 for all ϑ θ isin L2(FRd)
because we have chosen F0 ldquorich enoughrdquo Since our coefficients σ and b areLipschitz over Rd timesP2(R
d) this allows to get with the help of standard estimatesfor the SDEs for Xtξ and Xtxξ that for some constant C isin R+ only dependingon the Lipschitz constants of σ and b
E[
supsisin[tT ]
∣∣Xtξ1s minus Xtξ2
s
∣∣2]le CE
[|ξ1 minus ξ2|2]
(39)ξ1 ξ2 isin L2(
Ft Rd) t isin [0 T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
MEAN-FIELD SDES AND ASSOCIATED PDES 837
and for some Cp isin R depending only on p and the Lipschitz constants of thecoefficients
E[
supsisin[tv]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p](310)
le Cp
(|x1 minus x2|p +
int v
tW2(P
Xtξ1r
PX
tξ2r
)p dr
)
for all x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) 0 le t le v le T On the other hand from
the SDE for Xtxξ we derive easily that Xtξ primeξ (= Xtxξ |x=ξ prime) obeys the samelaw as Xtξ (= Xtxξ |x=ξ ) whenever ξ ξ prime isin L2(Ft Rd) have the same law Thisallows to deduce from the latter estimate for p = 2
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
le supsisin[tv]
E[∣∣Xtξ prime
1ξ1s minus X
tξ prime2ξ2
s
∣∣2] le E[
supsisin[tv]
∣∣Xtξ prime1ξ1
s minus Xtξ prime
2ξ2s
∣∣2](311)
le C
(E
[∣∣ξ prime1 minus ξ prime
2∣∣2] +
int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
) v isin [t T ]
for all ξ prime1 ξ
prime2 isin L2(Ft Rd) with Pξ prime
1= Pξ1 and Pξ prime
2= Pξ2 Hence taking at the
right-hand side of (311) the infimum over all such ξ prime1 ξ
prime2 isin L2(Ft Rd) and con-
sidering the above characterization of the 2-Wasserstein metric we get
supsisin[tv]
W2(PX
tξ1s
PX
tξ2s
)2
(312)
le C
(W2(Pξ1Pξ2)
2 +int v
tW2(P
Xtξ1r
PX
tξ2r
)2 dr
)
v isin [t T ] Then Gronwallrsquos inequality implies
supsisin[tT ]
W2(PX
tξ1s
PX
tξ2s
)2 le CW2(Pξ1Pξ2)2
(313)t isin [0 T ] ξ1 ξ2 isin L2(
Ft Rd)
which allows to deduce from the estimate (310)
E[
supsisin[tT ]
∣∣Xtx1ξ1s minus Xtx2ξ2
s
∣∣p]le Cp
(|x1 minus x2|p + W2(Pξ1Pξ2)p)
(314)
for all t isin [0 T ] x1 x2 isin Rd ξ1 ξ2 isin L2(Ft Rd) The proof is complete now
REMARK 31 An immediate consequence of the above Lemma 31 is thatgiven (t x) isin [0 T ] timesR
d the processes Xtxξ1 and Xtxξ2 are indistinguishable
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
838 BUCKDAHN LI PENG AND RAINER
whenever the laws of ξ1 isin L2(Ft Rd) and ξ2 isin L2(Ft Rd) are the same But thismeans that we can define
XtxPξ = Xtxξ (t x) isin [0 T ] timesRd ξ isin L2(
Ft Rd)(315)
and extending the notation introduced in the preceding section for functions torandom variables and processes we shall consider the lifted process X
txξs =
XtxPξs = X
txξs s isin [t T ] (t x) isin [0 T ] times R
d ξ isin L2(Ft Rd) However weprefer to continue to write Xtxξ and reserve the notation XtxPξ for an inde-pendent copy of XtxPξ which we will introduce later
4 First-order derivatives of XtxPξ Having now defined by the above rela-tion the process Xtxμ for all μ isin P2(R
d) the question of its differentiability withrespect to μ raises it will be studied through the Freacutechet differentiability of themapping L2(Ft Rd) ξ rarr X
txξs isin L2(FsRd) s isin [t T ] For this we suppose
the followingHypothesis (H1) The couple of coefficients (σ b) belongs to C
11b (Rd times
P2(Rd) rarr R
dtimesd times Rd) that is the components σij bj 1 le i j le d have the
following properties
(i) σij (x middot) bj (x middot) belong to C11b (P2(R
d)) for all x isin Rd
(ii) σij (middotμ) bj (middotμ) belong to C1b(Rd) for all μ isin P2(R
d)(iii) The derivatives partxσij partxbj Rd timesP2(R
d) rarr Rd and partμσij partμbj Rd times
P2(Rd) timesR
d rarr Rd are bounded and Lipschitz continuous
Before discussing the differentiability of Xtxμ with respect to the probabilitymeasure μ let us recall in a preparing step its L2-differentiability with respect to x
LEMMA 41 Let (t x) isin [0 T ] times Rd and ξ isin L2(Ft Rd) Under our above
Hypothesis (H1) the process XtxPξ = (XtxPξs )sisin[tT ] is L2-differentiable with
respect to x for all s isin [t T ] More precisely there is a (unique) processpartxX
txPξ isin S2([t T ]Rdtimesd) such that
E[
supsisin[tT ]
∣∣Xtx+hPξs minus X
txPξs minus partxX
txPξs h
∣∣2]= o
(|h|2) as Rd h rarr 0
[Recall that o(|h|) stands for an expression which tends quicker to zero than
h o(|h|)|h| rarr 0 as h rarr 0] Moreover partxXtxPξ = (partxi
XtxPξ
sj )1leijled isinS2([t T ]Rdtimesd) is the unique solution of the following SDE
partxiX
txPξ
sj = δij +dsum
k=1
int s
tpartxk
bj
(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dr
+dsum
k=1
int s
t(partxk
σj)(X
txPξr P
Xtξr
)partxi
XtxPξ
rk dBr (41)
s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
MEAN-FIELD SDES AND ASSOCIATED PDES 839
1 le i j le d (here δij denotes the Kronecker symbol it equals 1 if i = j andis equal to zero otherwise) Furthermore we have the following estimates for theprocess partxX
txPξ For every p ge 2 there is some constant Cp isin R such that forall t isin [0 T ] x xprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣partxXtxPξs
∣∣p]le Cp
(42)(ii) E
[sup
sisin[tT ]∣∣partxX
txPξs minus partxX
txprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
PROOF Considering for fixed (t ξ) the coefficients σ(s x) = σ(xPX
tξs
)
and b(s x) = b(xPX
tξs
) and taking into account that these coefficients are Lip-
schitz in x uniformly with respect to s we get the L2-differentiability of XtxPξ
and the SDE satisfied by its L2-derivative directly from the corresponding classi-cal result The proof for estimates for the L2-derivative combines standard SDEestimates with the argument developed in the proof for the estimates for XtxPξ
REMARK 41 From the above lemma we see that for given (t ξ) isin [0 T ]timesL2(Ft Rd) the process partxX
tξPξs = partxX
txPξs |x=ξ s isin [t T ] is the unique solu-
tion in S2([t T ]Rdtimesd) of the SDE
partxXtξPξs = I +
int s
t(partxσ )
(Xtξ
r PX
tξr
)partxX
tξPξr dBr
(43)+
int s
t(partxb)
(Xtξ
r PX
tξr
)partxX
tξPξr dr s isin [t T ]
Moreover it satisfies
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣p|Ft
]le sup
xisinRE
[sup
sisin[tT ]∣∣partxX
txPξs
∣∣p]le Cp P -as(44)
for some real constant Cp only depending on p ge 2 and the bounds of partxσ and partxb
Before giving the main statement of this section concerning the Freacutechet deriva-tive of L2(Ft Rd) ξ rarr Xtxξ = XtxPξ and thus of the differentiability of theprocess with respect to the probability law Pξ let us begin with a heuristic com-putation for the directional derivative Subsequently we will prove that it is indeedthe directional derivative and defines a Gacircteaux derivative which on its part isLipschitz and hence a Freacutechet derivative We will make the computations for di-mension d = 1 the case d ge 1 can be obtained by an easy extension
Let (t x) isin [0 T ] times R ξ isin L2(Ft )(= L2(Ft R)) and consider an arbitraryldquodirectionrdquo η isin L2(Ft ) Then supposing in a first attempt that the directionalderivative
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](45)
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-
840 BUCKDAHN LI PENG AND RAINER
exists we consider the SDE
Xtxξ+hηs = x +
int s
tσ
(X
txPξ+hηr P
Xtξ+hηr
)dBr
(46)+
int s
tb(X
txPξ+hηr P
Xtξ+hηr
)dr
s isin [t T ] which after lifting the process XtxPξ and the coefficients from thespace RtimesP2(R) to Rtimes L2(F) takes the form
Xtxξ+hηs = x +
int s
tσ
(Xtxξ+hη
r Xtξ+hηr
)dBr
(47)+
int s
tb(Xtxξ+hη
r Xtξ+hηr
)dr
s isin [t T ] and we derive formally this equation with respect to h at h = 0 For thiswe denote the Freacutechet derivatives of σ and b with respect to their second variableby Dϑ and we note that morally the L2-derivative of X
tξ+hηs is given by
parthXtξ+hηs |h=0 = partxX
txPξs |x=ξ middot η + Y txξ
s (η)|x=ξ (48)
Recalling that for some real C independent of (t xPξ )
E[
supsisin[tT ]
∣∣partxXtxP ξs
∣∣2]le C
we see that for partxXtξPξs = partxX
txPξs |x=ξ
E[
supsisin[tT ]
∣∣partxXtξPξs
∣∣2|Ft
]le C P -as(49)
Using the notation
Y tξs (η) = Y txξ
s (η)|x=ξ (410)
we get by formal differentiation of the above equation
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
s (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
s (η) dr
(411)+
int s
t(Dϑσ )
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dBr
+int s
t(Dϑ b)
(Xtxξ
r Xtξr
)(partxX
tξPξs η + Y tξ
s (η))dr s isin [t T ]
As concerns the above term (Dϑσ )(Xtxξr X
tξr )(partxX
tξPξs η + Y
tξs (η)) we see
from the definition of the differentiability of σ with respect to the probability mea-
MEAN-FIELD SDES AND ASSOCIATED PDES 841
sure that
(Dϑσ )(yXtξ
r
)(partxX
tξPξs η + Y tξ
s (η))
(412)= E
[(partμσ)
(yP
Xtξs
Xtξs
) middot (partxX
tξPξs η + Y tξ
s (η))]
y isinR
Now for given ξ η isin L2(Ft ) we denote by (ξ η B) a copy of (ξ ηB) on( F P ) while Xtξ is the solution of the SDE for Xtξ but driven by the Brow-
nian motion B and with initial value ξ Moreover XtxPξ denotes the solution of
the SDE for XtxPξ but governed by B instead of B
Xtξs = ξ +
int s
tσ
(Xtξ
r PX
tξr
)dBr +
int s
tb(Xtξ
r PX
tξr
)dr
XtxPξs = x +
int s
tσ
(X
txPξr P
Xtξr
)dBr +
int s
tb(X
txPξr P
Xtξr
)dr s isin [t T ]
Obviously XtxPξ = XtxPξ x isin R and (ξ η XtxPξ B) is an independent
copy of (ξ ηXtxPξ B) defined over ( F P ) Then due to the notation al-
ready introduced partxXtxPξr is the L2(P )-derivative of X
txPξr with respect to x
partxXtξ Pξr = partxX
txPξr |x=ξ and
Y txξs (η) = L2(P ) minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](413)
Having these notation in mind as well as the fact that the expectation E[middot] withrespect to P applies only to variables endowed with a tilde we see that
(Dϑσ )(Xtxξ
s Xtξr
) middot (partxX
tξPξs η + Y tξ
s (η))
= E[(partμσ)
(X
txPξs P
Xtξs
Xt ξ )s
) middot (partxX
tξ Pξs η + Y t ξ
s (η))]
(414)
s isin [t T ]Using also the corresponding formula for (Dϑb)(X
txξs X
tξr )(partxX
tξPξs η +
Ytξs (η)) and taking into account that partxσ (xϑ) = partxσ (xPϑ) and partxb(xϑ) =
partxb(xPϑ) (xϑ) isin Rtimes L2(F) we obtain
Y txξs (η) =
int s
t(partxσ )
(Xtxξ
r Xtξr
)Y txξ
r (η) dBr
+int s
t(partxb)
(Xtxξ
r Xtξr
)Y txξ
r (η) dr
(415)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
842 BUCKDAHN LI PENG AND RAINER
s isin [t T ] Finally recalling the definition of the process Y tξ (η) we see thatY tξ (η) solves the SDE
Y tξs (η) =
int s
t(partxσ )
(Xtξ
r Xtξr
)Y tξ
r (η) dBr +int s
t(partxb)
(Xtξ
r Xtξr
)Y tξ
r (η) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dBr(416)
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr η + Y t ξ
r (η))]
dr
s isin [t T ] We remark that thanks to the boundedness of the first-order derivativesof the coefficients σ and b the system formed of the both equations above hasa unique solution (Y txξ (η) Y tξ (η)) isin S2([t T ]R2) Moreover both processesare linear in η isin L2(Ft ) and a standard estimate shows that
E[
supsisin[tT ]
∣∣Y txξs (η)
∣∣2]le CE
[η2]
η isin L2(Ft )(417)
for some constant C independent of (t xPξ ) This shows in particular that
Ytxξs (middot) is a bounded linear operator from L2(Ft ) to L2(Fs)
Y txξs (middot) isin L
(L2(Ft )L
2(Fs)) s isin [t T ]
We are now able to show in a rigorous manner that our process Ytxξs (η) is the
directional derivative of Xtxξs in direction η
LEMMA 42 Under assumption (H1) we have for all (t xPξ ) isin [0 T ] timesRtimes L2(Ft )
Y txξs (η) = L2 minus lim
hrarr0
1
h
(Xtxξ+hη
s minus Xtxξs
) s isin [t T ](418)
that is Ytxξs (η) is the directional derivative of X
txξs in direction η isin L2(Ft )
PROOF The proof uses standard arguments Let us sketch it and without re-stricting the generality of the argument we suppose b = 0 First using the contin-uous differentiability of σ we see that
σ(Xtxξ+hη
s PX
tξ+hηs
) minus σ(Xtxξ
s PX
tξs
)=
int 1
0partλ
(σ
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
))dλ
(419)
+int 1
0partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))dλ
= αs(xh)(Xtxξ+hη
s minus Xtxξs
) + E[βs(xh)
(Xtξ+hη
s minus Xtξs
)]
MEAN-FIELD SDES AND ASSOCIATED PDES 843
where
αs(xh) =int 1
0(partxσ )
(Xtxξ
s + λ(Xtxξ+hη
s minus Xtxξs
)P
Xtξ+hηs
)dλ
βs(xh) =int 1
0(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))dλ
s isin [t T ] x h isinR are adapted uniformly bounded processes which are indepen-dent of Ft Indeed while for α(xh) the statement is obvious concerning β(xh)
we have for s isin [t T ]partλ
(σ
(Xtxξ
s PX
tξs +λ(X
tξ+hηs minusX
tξs )
))= (Dϑσ )
(Xtxξ
s Xtξs + λ
(Xtξ+hη
s minus Xtξs
))(Xtξ+hη
s minus Xtξs
)(420)
= E[(partμσ)
(X
txPξs P
Xtξs +λ(X
tξ+hηs minusX
tξs )
Xt ξs + λ
(Xtξ+hη
s minus Xtξs
))times (
Xtξ+hηs minus Xtξ
s
)]
Moreover using the Lipschitz continuity of partμσ we see that for some C isin R onlydepending on the Lipschitz constant of partμσ
E[∣∣βs(xh) minus (partμσ)
(X
txPξs P
Xtξs
Xt ξs
)∣∣2]le C
int 1
0
(W2(PX
tξs +λ(X
tξ+hηs minusX
tξs )
PX
tξs
)2 + E[∣∣Xtξ+hη
s minus Xtξs
∣∣2])dλ
le CE[∣∣Xtξ+hη
s minus Xtξs
∣∣2](421)
le CE[E
[∣∣XtyPξ+hηs minus X
typrimePξs
∣∣2]|y=ξ+hηyprime=ξ
]le CE
[[∣∣y minus yprime∣∣2 + W2(Pξ+hηPξ )2]|y=ξ+hηyprime=ξ
]le Ch2E
[η2]
s isin [t T ] x h isinR ξ η isin L2(Ft )
Similarly for partxσ from its Lipschitz continuity and our estimates for the processXtxPξ we have for all p ge 2 there is some constant Cp such that
E[∣∣αs(xh) minus (partxσ )
(X
txPξs P
Xtξs
)∣∣p]le Cp
(E
[∣∣XtxPξ+hηs minus X
txPξs
∣∣p] + W2(PXtξ+hηs
PX
tξs
)p)
(422)le CpW2(Pξ+hηPξ )
p + C|h|pE[η2]p2
le Cp|h|pE[η2]p2
s isin [t T ] x h isin R ξ η isin L2(Ft )
844 BUCKDAHN LI PENG AND RAINER
Recall that with the processes α(xh) and β(xh) the difference between
XtxPξ+hηs and X
txPξs writes for all s isin [t T ] as follows
XtxPξ+hηs minus X
txPξs =
int s
tαr(x h)
(X
txPξ+hηr minus XtxPξ
)dBr
(423)+
int s
tE
[βr(xh)
(Xtξ+hη
r minus Xtξr
)]dBr
Consequently using the equation for Y txξ (η) we have
XtxPξ+hηs minus X
txPξs minus hY
txPξs (η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)(X
txPξ+hηr minus X
txPξr minus hY txξ
r (η))dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
)(424)
times (Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η)))]
dBr
+ R1(s x h)
with
R1(s xh)
=int s
t
(αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)) middot (X
txPξ+hηr minus X
txPξr
)dBr
+int s
tE
[(βr(xh) minus (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]dBr
From Houmllderrsquos inequality (422) and our estimates for XtxPξ we obtain
E[∣∣αr(xh) minus (partxσ )
(X
txPξr P
Xtξr
)∣∣2∣∣XtxPξ+hηr minus X
txPξr
∣∣2](425)
le Ch2E[η2]
W2(Pξ+hηPξ )2 le Ch4E
[η2]2
and (421) yields
E[∣∣E[(
βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)) middot (Xtξ+hη
r minus Xtξr
)]∣∣2]le E
[E
[∣∣βr(h x) minus (partμσ)(X
txPξr P
Xtξr
Xt ξr
)∣∣2](426)
times E[∣∣Xtξ+hη
r minus Xtξr
∣∣2]]le Ch4E
[η2]2
r isin [t T ] h x isin R ξ η isin L2(Ft )
Consequently the remainder R1(s xh) can be estimated as follows
E[
supsisin[tT ]
∣∣R1(s xh)∣∣2]
le Ch4E[η2]2
h x isin R ξ η isin L2(Ft )
MEAN-FIELD SDES AND ASSOCIATED PDES 845
and using Gronwallrsquos inequality we obtain for a suitable constant C not depend-ing on (t x ξ η) that for all u isin [t T ]
supxisinR
E[
supsisin[tu]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2(427)
+ C
int u
tE
[E
[∣∣Xtξ+hηr minus Xtξ
r minus h(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]2]
dr
In order to conclude let us now estimate
E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξ Pξr η + Y t ξ
r (η))∣∣]
= E[∣∣Xtξ+hη
r minus Xtξr minus h
(partxX
tξPξr η + Y tξ
r (η))∣∣]
For this we observe that
Xtξ+hηr minus Xtξ
r minus h(partxX
tξPξr η + Y tξ
r (η)) = I1(r h) + I2(r h)
where I1(r h) = (XtxPξ+hηr minus X
txPξr minus hY
txξr (η))|x=ξ satisfies
E[∣∣I1(r h)
∣∣]2 le supxisinR
E[∣∣XtxPξ+hη
r minus XtxPξr minus hY txξ
r (η)∣∣2]
and for I2(r h) = Xtξ+hηPξ+hηr minus X
tξPξ+hηr minus hpartxX
tξPξr η we have
E[∣∣I2(r h)
∣∣]2 le h2E[η2] int 1
0E
[∣∣partxXtξ+λhηPξ+hηr minus partxX
tξPξr
∣∣2]dλ
le h2E[η2] int 1
0E
[E
[∣∣partxXtyPξ+hηr minus partxX
typrimePξr
∣∣2]|y=ξ+λhηyprime=ξ
]dλ
le Ch4E[η2]2
Therefore combining these three latter estimates we obtain
E[
supsisin[tT ]
∣∣XtxPξ+hηs minus X
txPξs minus hY txξ
s (η)∣∣2]
le Ch4E[η2]2
for all h isin R η isin L2(Ft ) for some constant C independent of t isin [0 T ] x isin R
and ξ isin L2(Ft ) The proof is complete now
PROPOSITION 41 For any given t isin [0 T ] ξ isin L2(Ft ) x y isin R let(Utxξ (y)Utξ (y)
) = (Utxξ
s (y)Utξs (y)
)sisin[tT ] isin S
([t T ]R2)(428)
846 BUCKDAHN LI PENG AND RAINER
be the unique solution of the system of (uncoupled) SDEs
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)Utxξ
r (y) dr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
(429)+ (partμσ)
(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)Utξ
r (y) dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
(430)+ (partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
+ (partμb)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dr
s isin [t T ] where (U tξ (y) B) is supposed to follow under P exactly the samelaw as (Utξ (y)B) under P Then for all η isin L2(Ft ) the directional derivative
Ytxξs (η) of ξ rarr X
txξs = X
txPξs satisfies
Y txξs (η) = E
[Utxξ
s (ξ ) middot η] s isin [t T ] P -as(431)
REMARK 42 (1) One can consider U t ξ (y) as the unique solution of the SDEfor Utξ (y) but with the data (ξ B) instead of (ξB)
(2) Since the derivatives partxσ partxb partμσ and partμb are bounded and the processpartxX
tyPξ is bounded in L2 by a constant independent of y isin R it is easy to provethe existence of the solution Utξ (y) for the above SDE (430) and to show thatit is bounded in L2 by a constant independent of y isin R Once having the processUtξ (y) the existence of the solution Utxξ (y) of (429) is immediate
Before proving the above proposition let us state the following lemma
MEAN-FIELD SDES AND ASSOCIATED PDES 847
LEMMA 43 Assume (H1) Then for all p ge 2 there is some constant Cp isin R
such that for all t isin [0 T ] x xprime y yprime isin Rd and ξ ξ prime isin L2(Ft Rd)
(i) E[
supsisin[tT ]
∣∣Utxξs (y)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p]
(432)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
REMARK 43 From estimate (ii) of the above lemma we see that Utxξ (y)
depends on ξ isin L2(Ft ) only through its law Pξ This allows in analogy to XtxPξ to write UtxPξ (y) = Utxξ (y)
Moreover it is easy to verify that the solution UtxPξ (y) is (σ Br minus Bt r isin[t s] orNP )-adapted and hence independent of Ft and in particular of ξ Sub-stituting in UtxPξ (y) the random variable ξ for x we deduce from the uniquenessof the solution of the equation for Utξ (y) that
Utξs (y) = U
txPξs (y)|x=ξ s isin [t T ] P -as(433)
The same argument allows also to substitute Ft -measurable random variables fory in U
tξs (y)
PROOF OF LEMMA 43 We continue to restrict ourselves to the one-dimensional case d = 1 and to simplify a bit more we suppose also that b = 0By standard arguments already used in the proof of the estimates for XtxPξ which we combine with our estimates for XtxPξ and partxX
txPξ we see that forall t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
supsisin[tT ]
(∣∣Utxξs (y)
∣∣p + ∣∣Utξs (y)
∣∣p)] le Cp(434)
As concern the proof of the estimate
E[
supsisin[tT ]
(∣∣Utxξs (y) minus Utxprimeξ prime
s
(yprime)∣∣p + ∣∣Utξ
s (y) minus Utξ primes
(yprime)∣∣p)]
(435) le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
we notice that its central ingredient is the following estimate which uses the Lip-schitz property of partμσ with respect to all its variables as well as the boundednessof partμσ
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](436)
le Cp
(E
[∣∣XtxPξr minus X
txprimePξ primer
∣∣2p + (E
[∣∣XtyPξr minus X
typrimePξ primer
∣∣2])p]+ W2(PX
tξr
PX
tξ primer
)2p)12 + Cp
(E
[∣∣partxXtyPξs minus partxX
typrimePξ primes
∣∣2])p2
848 BUCKDAHN LI PENG AND RAINER
Once having the above estimate we can use our estimates for XtxPξ andpartxX
txPξ in order to deduce that
E[∣∣E[
(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(X
txprimePξ primer P
Xtξ primer
XtyprimePξ primer
) middot partxXtyprimePξ primer
]∣∣p](437)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
and
E[∣∣E[
(partμσ)(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr
minus (partμσ)(Xtξ prime
r PX
tξ primer
Xtξ primer
) middot partxXtyprimePξ primer
]∣∣p](438)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Consequently from the equations for Utxξ (y)Utxprimeξ prime(yprime) the above estimates
and Gronwallrsquos inequality we see that for all p ge 2 there is some constant Cp
such that for all t isin [0 T ] x xprime y yprime isin R and ξ ξ prime isin L2(Ft )
E[
suprisin[ts]
∣∣Utxξr (y) minus Utxprimeξ prime
r
(yprime)∣∣p]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
(439)
+ E
[int s
t
∣∣E[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]∣∣p2
dr
] s isin [t T ]
Then from this estimate for p = 2
E[
suprisin[ts]
∣∣Utξr (y) minus Utξ prime
r
(yprime)∣∣2]
= E[E
[sup
risin[ts]∣∣Utxξ
r (y) minus Utxprimeξ primer
(yprime)∣∣2]
|x=ξxprime=ξ prime]
(440)le C
(E
[∣∣ξ minus ξ prime∣∣2] + ∣∣y minus yprime∣∣2)+ E
[int s
tE
[∣∣U t ξr (y) minus U t ξ prime
r
(yprime)∣∣2]
dr
]
s isin [t T ] Applying Gronwallrsquos inequality to (440) and substituting the obtainedrelation in (439) we obtain (ii) of the lemma This completes the proof
We are now able to give the proof of Proposition 41
PROOF PROPOSITION 41 Let now (ξ η B) be a copy of (ξ ηB) indepen-dent of (ξ ηB) and (ξ η B) and defined over a new probability space ( F P )
MEAN-FIELD SDES AND ASSOCIATED PDES 849
which is different from (FP ) and ( F P ) the expectation E[middot] applies onlyto random variables over ( F P ) This extension to (ξ η E[middot]) here is done inthe same spirit as that from (ξ ηB) to (ξ η B)
In order to obtain the wished relation we substitute in the equation for Utξ (y)
for y the random variable ξ and we multiply both sides of the such obtained equa-tion by η [observe that ξ and η are independent of all terms in the equation forUtξ (y)] Then we take the expectation E[middot] on both sides of the new equationThis yields
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dBr(441)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dBr
+int s
tE
[E
[(partμb)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
dr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot E[U t ξ
r (ξ ) middot η]]dr s isin [t T ]
Taking into account that (ξ η) is independent of (ξ ηB) and (ξ η B) and of thesame law under P as (ξ η) under P we see that
E[E
[(partμσ)
(Xtξ
r PX
tξr
Xt ξ Pξr
) middot partxXtξ Pξr middot η]]
= E[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot partxXtξ Pξr middot η]
and the same relation also holds true for partμb instead of partμσ Thus the aboveequation takes the form
E[Utξ
s (ξ ) middot η]=
int s
tpartxσ
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dBr
+int s
tpartxb
(Xtξ
r PX
tξr
)E
[Utξ
r (ξ ) middot η]dr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η + E
[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμb)
(Xtξ
r PX
tξr
Xt ξr
) middot (partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dr s isin [t T ]
850 BUCKDAHN LI PENG AND RAINER
But this latter SDE is just equation (416) for Y tξ (η) and from the uniqueness ofthe solution of this equation it follows that
Y tξs (η) = E
[Utξ
s (ξ ) middot η] s isin [t T ](442)
Finally we substitute ξ for y in the SDE for UtxPξ (y) we multiply both sides ofthe such obtained equation by η and take after the expectation E[middot] at both sidesof the relation Using the results of the above discussion we see that this yields
E[U
txPξs (ξ ) middot η]=
int s
tpartxσ
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tpartxb
(X
txPξr P
Xtξr
)E
[U
txPξr (ξ ) middot η]
dr
(443)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dBr
+int s
tE
[(partμb)
(X
txPξr P
Xtξr
Xt ξr
) middot (partxX
tξ Pξr middot η + Y t ξ
r (η))]
dr
s isin [t T ]But this latter SDE is just that (415) satisfied by Y txPξ (η) Therefore from theuniqueness of the solution of this SDE it follows that
YtxPξs (η) = E
[U
txPξs (ξ ) middot η]
s isin [t T ] η isin L2(Ft )(444)
The proof is complete now
The preceding both statements allow to derive our main result We first derive itfor the one-dimensional case before stating it for the general case
PROPOSITION 42 Under the assumption (H1) for any (t x) isin [0 T ] timesR s isin [t T ] the mapping
L2(Ft ) ξ rarr Xtxξs = X
txPξs isin L2(Fs)
is Freacutechet differentiable Its Freacutechet derivative DξXtxξs satisfies
DξXtxξs (η) = Y txξ
s (η) = E[U
txPξs (ξ ) middot η]
η isin L2(Ft )(445)
REMARK 44 We observe that the latter relation satisfied by DξXtxξs (η) ex-
tends that for the derivative of deterministic functions with respect to a probabilitylaw to stochastic processes In this sense it is natural to define the derivative of
XtxPξs with respect to the probability law Pξ by putting
partμXtxPξs (y) = U
txPξs (y) s isin [t T ] x y isinR
MEAN-FIELD SDES AND ASSOCIATED PDES 851
We observe that with this definition for any (t x) isin [0 T ] timesR s isin [t T ]DξX
txξs (η) = Y txξ
s (η) = E[partμX
txPξs (ξ ) middot η]
η isin L2(Ft )(446)
PROOF OF PROPOSITION 42 Let (t x) isin [0 T ] times R ξ isin L2(Ft ) ands isin [t T ] We recall that the directional derivative Y
txξs (η) of L2(Ft ) ξ rarr
Xtxξs isin L2(Fs) in direction η isin L2(Fs) has the property that Y
txξs (middot) isin
L(L2(Ft )L2(Fs)) Let us denote the operator norm middot L(L2L2) in L(L2(Ft )
L2(Fs)) Then using the preceding lemma since
E[∣∣Y txξ
s (η)∣∣2] = E
[∣∣E[U
txPξs (ξ ) middot η]∣∣2]
le E[E
[U
txPξs (ξ )2]
E[η2]] le C2E
[η2]
η isin L2(Ft )
for some positive constant C depending only on the coefficients b and σ we have∥∥Y txξs
∥∥2L(L2L2) = sup
E
[∣∣Y txξs (η)
∣∣2] η isin L2(Ft ) with E[η2] le 1
le C
Moreover for all (t x) isin [0 T ] timesR ξ ξ prime isin L2(Ft ) s isin [t T ]
E[∣∣Y txξ
s (η) minus Y txξ primes (η)
∣∣2] = E[∣∣E[(
UtxPξs (ξ ) minus U
txPξ primes (ξ )
) middot η]∣∣2]le E
[E
[∣∣UtxPξs (ξ ) minus U
txPξ primes (ξ )
∣∣2]E
[η2]]
le C2W2(Pξ Pξ prime)2E[η2]
η isin L2(Ft )
that is∥∥Y txξs minus Y txξ prime
s
∥∥2L(L2L2)
= supE
[∣∣Y txξs (η) minus Y txξ prime
s (η)∣∣2] η isin L2(Ft ) with E[η2] le 1
le CW2(Pξ Pξ prime)2 le CE
[∣∣ξ minus ξ prime∣∣2] ξ ξ prime isin L2(Ft )
This proves that the directional derivative Ytxξs is a bounded operator (and hence
a Gacircteaux derivative) which moreover is continuous in ξ which proves that it iseven the Freacutechet derivative of X
txξs with respect to ξ The proof is complete
A direct generalization of our preceding computations from the one-dimen-sional to the multi-dimensional case allows to establish the following general re-sult
THEOREM 41 Let (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) satisfyassumption (H1) Then for all 0 le t le s le T and x isin R
d the mapping
852 BUCKDAHN LI PENG AND RAINER
L2(Ft Rd) ξ rarr Xtxξs = X
txPξs isin L2(FsRd) is Freacutechet differentiable with
Freacutechet derivative
DξXtxξs (η) = E
[U
txPξs (ξ ) middot η] =
(E
[dsum
j=1
UtxPξ
sij (ξ ) middot ηj
])1leiled
(447)
for all η = (η1 ηd) isin L2(Ft Rd) where for all y isin Rd UtxPξ (y) =
((UtxPξ
sij (y))sisin[tT ])1leijled isin S2F(t T Rdtimesd) is the unique solution of the SDE
UtxPξ
sij (y) =dsum
k=1
int s
tpartxk
σi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dBr
+dsum
k=1
int s
tpartxk
bi
(X
txPξr P
Xtξr
) middot UtxPξ
rkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(448)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
txPξr
dr
s isin [t T ]1 le i j le d and Utξ (y) = ((Utξsij (y))sisin[tT ])1leijled isin S2
F(t T
Rdtimesd) is that of the SDE
Utξsij (y) =
dsumk=1
int s
tpartxk
σi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dB
r
+dsum
k=1
int s
tpartxk
bi
(Xtξ
r PX
tξr
) middot Utξrkj (y) dr
+dsum
k=1
int s
tE
[(partμσi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
(449)+ (partμσi)k
(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dBr
+dsum
k=1
int s
tE
[(partμbi)k
(zP
Xtξr
XtyPξr
) middot partxjX
tyPξ
rk
+ (partμbi)k(zP
Xtξr
Xtξr
) middot Utξrkj (y)
]∣∣∣z=X
tξr
dr
s isin [t T ]1 le i j le d
MEAN-FIELD SDES AND ASSOCIATED PDES 853
REMARK 45 As for the one-dimensional case following the definition ofthe derivative of a function f P2(R
d) rarr R explained in Section 2 we con-
sider UtxPξs (y) = (U
txPξ
sij (y))1leijled as derivative over P2(Rd) of X
txPξs =
(XtxPξ
si )1leiled with respect to Pξ As notation for this derivative we use that al-ready introduced for functions s isin [t T ]
partμXtxPξs (y) = (
partμXtxPξ
sij (y))1leijled = U
txPξs (y)
(450)= (
UtxPξ
sij (y))1leijled
5 Second-order derivatives of XtxPξ Let us come now to the study ofthe second-order derivatives of the process XtxPξ For this we shall suppose thefollowing Hypothesis for the remaining part of the paper
Hypothesis (H2) Let (σ b) belong to C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) that
is (σ b) isin C11b (Rd times P2(R
d) rarr Rdtimesd times R
d) [see Hypothesis (H1)] and thederivatives of the components σij bj 1 le i j le d have the following proper-ties
(i) partkσij (middot middot) partkbj (middot middot) belong to C11(Rd timesP2(Rd)) for all 1 le k le d
(ii) partμσij (middot middot middot) partμbj (middot middot middot) belong to C11(Rd timesP2(Rd) timesR
d)(iii) All the derivatives of σij bj up to order 2 are bounded and Lipschitz
REMARK 51 With the existence of the second-order mixed derivativespartxl
(partμσij (xμ y)) and partμ(partxlσij (xμ))(y) (xμ y) isin R
d timesP2(Rd) timesR
d thequestion of their equality raises Indeed under Hypothesis (H2) they coincideand the same holds true for those for b More precisely we have the followingstatement
LEMMA 51 Let g isin C21b (Rd timesP2(R
d) rarrRdtimesd timesR
d) [in the sense of Hy-pothesis (H2)] Then for all 1 le l le d
partxl
(partμg(xμy)
) = partμ
(partxl
g(xμ))(y) (xμ y) isin R
d timesP2(R
d) timesRd
PROOF Let us restrict to d = 1 Following the argument of Clairautrsquos theo-rem we have for all (x ξ) (z η) isin Rtimes L2(F)
I = (g(x + zPξ+η) minus g(x + zPξ )
) minus (g(xPξ+η) minus g(xPξ )
)=
int 1
0E
[((partμg)(x + zPξ+sη ξ + sη) minus (partμg)(xPξ+sη ξ + sη)
)η]ds
=int 1
0
int 1
0E
[partx(partμg)(x + tzPξ+sη ξ + sη)ηz
]ds dt
= E[partx(partμg)(xPξ ξ)ηz
] + R1((x ξ) (z η)
)
854 BUCKDAHN LI PENG AND RAINER
with |R1((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) and at the same time
I = (g(x + zPξ+η) minus g(xPξ+η)
) minus (g(x + zPξ ) minus g(xPξ )
)=
int 1
0
((partxg)(x + tzPξ+η) minus (partxg)(x + tzPξ )
)dt middot z
=int 1
0
int 1
0E
[partμ
((partxg)(x + tzPξ+sη)
)(ξ + sη)η
]ds dt middot z
= E[partμ
((partxg)(xPξ )
)(ξ)η
]z + R2
((x ξ) (z η)
)
with |R2((x ξ) (z η))| le C(|η|L2 middot |z|2 + |η|2L2 middot |z|) where C is the Lips-
chitz constant of partμ(partxg) and partx(partμg) It follows that partx(partμg)(xPξ ξ) =partμ((partxg)(xPξ ))(ξ) P -as and hence
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) Pξ (dy)-ae
Letting ε gt 0 and θ be a standard normally distributed random variable whichis independent of ξ and taking ξ + εθ instead of ξ we have
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) Pξ+εθ (dy)-ae
and thus dy-as on R Taking into account that partx(partμg) partμ(partxg) are Lipschitzthis yields
partx(partμg)(xPξ+εθ y) = partμ
((partxg)(xPξ+εθ )
)(y) for all y isin R
Finally using W2(Pξ+εθ Pξ ) le ε(E[θ2])12 = Cε and again the Lipschitz prop-erty of partx(partμg) and partμ(partxg) we obtain by taking the limit as ε darr 0
partx(partμg)(xPξ y) = partμ
((partxg)(xPξ )
)(y) for all x y isinR ξ isin L2(F)
The proof is complete
After the above preparing discussion let us study now the second-order deriva-tives Following the approach for the first-order derivatives we restrict here our-selves to the one-dimensional and to shorten the formulas let us put b = 0 Weemphasize that the general case with dimension d ge 1 and a drift b not identicallyequal to zero can be obtained with a straight-forward extension For the purpose ofbetter comprehension the main result in this section concerning the general casewill be given only after our computations
We begin with recalling that the process partxXtxPξ isin S([t T ]R) is the unique
solution of the SDE
partxXtxPξs = 1 +
int s
t(partxσ )
(X
txPξr P
Xtξr
)partxX
txPξr dBr s isin [t T ](51)
(Recall that b = 0 in our computations) With the same classical arguments whichhave shown the existence of this first-order derivative in L2-sense with respectto x we can prove the existence of the second-order L2-derivative part2
xXtxPξ withrespect to x and we can characterize it as the unique solution in S([t T ]R) of
MEAN-FIELD SDES AND ASSOCIATED PDES 855
the SDE
part2xX
txPξs =
int s
t
((partxσ )
(X
txPξr P
Xtξr
)part2xX
txPξr
(52)+ (
part2xσ
)(X
txPξr P
Xtξr
)(partxX
txPξr
)2)dBr s isin [t T ]
We also notice that with standard arguments we obtain the following lemma
LEMMA 52 Under Hypothesis (H2) for all p ge 2 there is some con-stant Cp only depending on p and the coefficients σ and b such that for allt isin [0 T ] x xprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣part2xX
txPξs
∣∣p]le Cp
(53)(ii) E
[sup
sisin[tT ]∣∣part2
xXtxPξs minus part2
xXtxprimePξ primes
∣∣p]le Cp
(∣∣x minus xprime∣∣p + W2(Pξ Pξ prime)p)
In the same manner as one proves the L2-differentiability of x rarr XtxPξs
s isin [t T ] one proves that for ξ rarr partμXtxPξs (y) and y rarr partμX
txPξs (y) s isin [t T ]
Standard arguments give the following result
LEMMA 53 Under Hypothesis (H2) for all t isin [0 T ] ξ isin L2(Ft ) theprocess partμXtxPξ (y) is L2-differentiable in x y isin R and its derivativespartx(partμXtxPξ (y)) party(partμXtxPξ (y)) isin S2([t T ]R) are the unique solutions ofthe SDE
partx
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)partx
(partμX
txPξr (y)
)dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)partμX
txPξr (y) middot partxX
txPξr dBr
(54)+
int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
+ partx(partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]partxX
txPξr dBr
and
party
(partμXtxPξ (y)
) =int s
t(partxσ )
(X
txPξr P
Xtξr
)party
(partμX
txPξr (y)
)dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot (partxX
tyPξr
)2]dBr
(55)+
int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr
856 BUCKDAHN LI PENG AND RAINER
respectively where the latter equation is coupled with the SDE
partyUtξs (y) =
int s
t(partxσ )
(Xtξ
r PX
tξr
)partyU
tξr (y) dBr
+int s
tE
[party(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)times (
partxXtyPξr
)2]dBr(56)
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
)part2x X
tyPξr
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)partyU
tξr (y)
]dBr s isin [t T ]
which solution partyUtξ (y) isin S([t T ]R) is the L2-derivative with respect to y isinR
of Utξ (y) = partμXtxPξ (y)|x=ξ Moreover the techniques of estimate explained inthe preceding section yield that for all p ge 2 there is some Cp isin R such that for
all t isin [0 T ] x xprime y yprime isinR ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
)∣∣p + ∣∣party
(partμXtxPξ (y)
)∣∣p)] le Cp
(ii) E[
supsisin[tT ]
(∣∣partx
(partμXtxPξ (y)
) minus partx
(partμXtxprimePξ prime (yprime))∣∣p
(57)+ ∣∣party
(partμXtxPξ (y)
) minus party
(partμXtxprimePξ prime (yprime))∣∣p)]
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
Let us now consider the derivative of the process partxXtxPξ with respect to the
probability law Knowing already that the mixed second-order derivatives partxpartμ
and partμpartx coincide for all functions from C11(Rd timesP2(R)) we guess the follow-ing
LEMMA 54 Under Hypothesis (H2) for all (t x) isin [0 T ]timesR ξ isin L2(Ft )
partμpartxXtxPξs (y) = partxpartμX
txPξs (y) s isin [t T ]
PROOF Recall that we have put b = 0 Then using the fact that partxσ and part2xσ
are bounded a standard estimate involving the SDEs for XtxPξ and partxXtxPξ
shows that there is some constant C isin R such that for all (t x) isin [0 T ] timesR ξ isinL2(Ft ) and all s isin [t T ] h isin R 0
E
[∣∣∣∣1
h
(X
tx+hPξs minus X
txPξs
) minus partxXtxPξs
∣∣∣∣2]le Ch2(58)
MEAN-FIELD SDES AND ASSOCIATED PDES 857
Then for all direction η isin L2(Ft ) and all λ isin R 0 we have for arbitrary h isinR 0
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
((X
tx+hPξ+ληs minus X
tx+hPξs
) minus (X
txPξ+ληs minus X
txPξs
))minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0E
[(partμX
tx+hPξ+vηs (ξ + vη)
minus partμXtxPξ+vηs (ξ + vη)
) middot η]dv minus E
[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[partx
(partμX
tx+uPξ+vηs (ξ + vη)
) middot η]dudv
minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2]
le Ch2
λ2 + E
[∣∣∣∣ 1
hλ
int λ
0
int h
0E
[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
)minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η∣∣]dudv
∣∣∣∣2]
Thus taking into account that
E[∣∣partx
(partμX
tx+uPξ+vηs (ξ + vη)
) minus partx
(partμX
txPξs (ξ )
)∣∣ middot |η|](59)
le C(|u| + W2(Pξ+vηPξ )
)E
[η2]12 + C|v|E[
η2]
we obtain
E
[∣∣∣∣1
λ
(partxX
txPξ+ληs minus partxX
txPξs
) minus E[partx
(partμX
txPξs (ξ )
) middot η]∣∣∣∣2](510)
le C
(h2
λ2 + λ2E[η2]2
)minusrarr Cλ2E
[η2]2
as h rarr 0
But this proves that E[partx(partμXtxPξs (ξ )) middot η] is the directional derivative of
partxXtxPξ in direction η Using the SDE for partx(partμXtxPξ (y)) we show like in
the preceding section that this directional derivative is in fact a Freacutechet derivative
Dξ
(partxX
txξ )(η) = E
[partx
(partμX
txPξs (ξ )
) middot η]
858 BUCKDAHN LI PENG AND RAINER
of the lifted mapping L2(Ft ) ξ rarr partxXtxξs = partxX
txPξs s isin [t T ] The state-
ment of the lemma follows directly
It remains to study the second-order derivative of XtxPξ with respect to theprobability law that is the derivative of partμXtxPξ (y) in Pξ Recall that usingthat b = 0 we have that partμXtxPξ (y) = UtxPξ (y) is the unique solution inS2([t T ]R) of the following (uncoupled) SDE
Utxξs (y) =
int s
tpartxσ
(X
txPξr P
Xtξr
)Utxξ
r (y) dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr(511)
+ (partμσ)(X
txPξr P
Xtξr
Xt ξr
)U t ξ
r (y)]dBr
Utξs (y) =
int s
tpartxσ
(Xtξ
r PX
tξr
)Utξ
r (y) dBr
+int s
tE
[(partμσ)
(Xtξ
r PX
tξr
XtyPξr
) middot partxXtyPξr(512)
+ (partμσ)(Xtξ
r PX
tξr
Xt ξr
) middot U t ξr (y)
]dBr
Arguing similarly as in the preceding section in the proof of the Freacutechet differ-entiability of the lifted process Xtxξ = XtxPξ we derive formally the lifted
process Utxξs (y) = U
txPξs (y) for fixed (t x) isin [0 T ] times R y isin R in direction
η isin L2(Ft ) This gives for the formal directional derivative
Ztxξs (y η) = L2 minus lim
hrarr0
1
h
(Utxξ+hη
s (y) minus Utxξs (y)
) s isin [t T ]
the following SDE
Ztxξs (y η)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)Ztxξ
r (y η) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y)E
[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)
(partxX
tξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[partμpartxX
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
]
MEAN-FIELD SDES AND ASSOCIATED PDES 859
times E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr(513)
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
times E[U
tyPξr (ξ ) middot η]]
dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot (partx
(partμX
tξ Pξr (y)
) middot η
+ Zt ξr (y η)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot E[U
txPξr (ξ ) middot η]
dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y)
(partxX
tξPξ
r middot η
+ E[U
tξ
r (ξ ) middot η])]]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot (
partxXtξ Pξr middot η
+ E[U t ξ
r (ξ ) middot η])]dBr
s isin [t T ] where Ztξ (y η) = ZtxPξ (y η)|x=ξ isin S2([t T ]R) is the unique so-lution of the above equation after substituting everywhere x = ξ Let us also point
out that in the above equation we have used the notation (Xtξ
Utξ
(y)) it is usedin the same sense as the corresponding processes endowed with ˜ or We con-sider a copy (ξ ηB) independent of (ξ ηB) (ξ η B) and (ξ η B) and the
process Xtξ
is the solution of the SDE for Xtξ and Utξ
that of the SDE for Utξ but both with the data (ξB) instead of (ξB)
Let us comment also the expression part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z)
in the above formula Recalling that part2μσ(xPϑ y z) = partμ(partμσ(xPϑ y))(z) is
defined through the relation Dϑ [partμσ (xϑ y)](θ) = E[part2μσ(xPϑ yϑ) middot θ ] for
ϑ θ isin L2(F) x y isin R where Dϑ denotes the Freacutechet derivative with respect toϑ we have namely for the Freacutechet derivative of L2(F) ϑ rarr partμσ (xϑ y) =(partμσ)(xPϑ y) in direction θ isin L2(F)
Dϑ
[partμσ (xϑ y)
](θ) = E
[(part2μσ
)(xPϑ yϑ) middot θ]
860 BUCKDAHN LI PENG AND RAINER
Then of course the Freacutechet derivative of ξ rarr partμσ (XtxPξr X
tξr XtyPξ ) is
given by
Dξ
[partμσ
(X
txPξr Xtξ
r XtyPξ)]
(η)
= partx(partμσ)(X
txPξr P
Xtξr
XtyPξr
) middot E[U
txPξr (ξ ) middot η]
+ E[(
part2μσ
)(X
txPξr P
Xtξr
XtyPξ Xtξ
r
)(514)
times (partxX
tξPξ
r middot η + E[U
tξ
r (ξ ) middot η])]+ party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot E[U
tyPξr (ξ ) middot η]
η isin L2(Ft )
r isin [t T ] but this is just what has been used for the above formula in combinationwith arguments already developed in the preceding section
Let us now compare the solution Ztxξ (y η) of the above SDE with the processUtxPξ (y z) isin S2
F(t T ) defined as the unique solution of the following SDE
UtxPξs (y z)
=int s
t(partxσ )
(X
txPξr P
Xtξr
)U
txPξr (y z) dBr
+int s
t
(part2xσ
)(X
txPξr P
Xtξr
)U
txPξr (y) middot UtxPξ
r (z) dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
XtzPξr
)U
txPξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[partμ(partxσ )
(X
txPξr P
Xtξr
Xt ξr
)U
txPξr (y)U tξ
r (z)]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partμ
(partxX
tyPξr
)(z)
]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tzPξ
r
)times partxX
tyPξr middot partxX
tzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
XtyPξr X
tξ
r
) middot partxXtyPξr U
tξ
r (z)]]
dBr
(515)+
int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtyPξr
) middot partxXtyPξr U
tyPξr (z)
]dBr
MEAN-FIELD SDES AND ASSOCIATED PDES 861
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y z)
]dBr
+int s
tE
[(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot partx
(partμX
tzPξr (y)
)]dBr
+int s
tE
[partx(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y)
] middot UtxPξr (z) dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tzPξ
r
)times U t ξ
r (y) middot partxXtzPξ
r
]]dBr
+int s
tE
[E
[(part2μσ
)(X
txPξr P
Xtξr
Xt ξr X
tξ
r
) middot U t ξr (y) middot Utξ
r (z)]]
dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
XtzPξr
) middot U t ξr (y) middot partxX
tzPξr
]dBr
+int s
tE
[party(partμσ)
(X
txPξr P
Xtξr
Xt ξr
) middot U t ξr (y) middot U t ξ
r (z)]dBr
s isin [0 T ]combined with the SDE for Utξ (y z) = (U
tξs (y z))sisin[tT ] obtained by sub-
stituting x = ξ in the equation for UtxPξ (y z) (recall namely that Xtξ =XtxPξ |x=ξ U
tξ = UtxPξ |x=ξ ) We consider now the processes E[UtxPξ (y ξ ) middotη] and E[Utξ (y ξ ) middot η] Substituting first z = ξ in the SDE for UtxPξ (y z) andthat for Utξ (y z) then multiplying the both sides of these SDEs with η and taking
the expectation E[middot] of this product we get just the SDEs solved by ZtxPξs (y η)
and Ztξs (y η) (see also the corresponding proof for the first-order derivatives in
the preceding section) and from the uniqueness of the solution of these SDEs weconclude that
Ztxξs (y η) = E
[U
txPξs (y ξ ) middot η]
s isin [t T ] y isin R(516)
We also observe that the SDEs for UtxPξ (y z) and Utξ (y z) allow to make thefollowing estimates
LEMMA 55 Under Hypothesis (H2) for all p ge 2 there is some constantCp isinR such that for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft )
(i) E[
supsisin[tT ]
∣∣UtxPξs (y z)
∣∣p]le Cp
(ii) E[
supsisin[tT ]
∣∣UtxPξs (y z) minus U
txprimePξ primes
(yprime zprime)∣∣p]
(517)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
862 BUCKDAHN LI PENG AND RAINER
The estimates of Lemma 55 allow to show in analogy to the argument devel-
oped for the proof of the Freacutechet differentiability of ξ rarr Xtxξs = X
txPξs that
the mappings L2(Ft ) ξ rarr UtxPξs (y) isin L2(Fs) and L2(Ft ) ξ rarr U
tξs (y) isin
L2(Fs) are Freacutechet differentiable and
Dξ
[partμX
txPξs (y)
](η) = Dξ
[U
txPξs (y)
](η) = E
[U
txPξs (y ξ ) middot η]
Dξ
[partμXtξ
s (y)](η) = Dξ
[Utξ
s (y)](η)
= partxUtxPξs (y)|x=ξ middot η + E
[Utξ
s (y ξ ) middot η]
But this means that
part2μX
txPξs (y z) = U
txPξs (y z) s isin [0 T ](518)
for all t isin [0 T ] x y z isin R ξ isin L2(Ft )Let us now summarize the above results concerning the second-order derivatives
and formulate the main result concerning them but for dimension d ge 1 and b notnecessarily equal to zero It can be proved by a straight forward extension of thepreceding computations
THEOREM 51 Under Hypothesis (H2) the first-order derivatives partxiX
txPξs
1 le i le d and partμXtxPξs (y) are in L2-sense differentiable with respect to x and
y and interpreted as functional of ξ isin L2(Ft Rd) they are also Freacutechet dif-ferentiable with respect to ξ Moreover for all t isin [0 T ] x y z isin R and ξ isinL2(Ft Rd) there are stochastic processes partμ(partxi
XtxPξ )(y) isin S2([t T ]Rd)1 le i le d and part2
μXtxPξ (y z) = partμ(partμXtxPξ (y))(z) in S2([t T ]Rdtimesd) such
that for all η isin L2(Ft Rd) the Freacutechet derivatives in ξ Dξ [partxXtxPξs ](middot) and
Dξ [partμXtxPξs (y)](middot) satisfy
(i) Dξ
[partxi
XtxPξs
](η) = E
[partμ
(partxi
XtxPξs
)(ξ ) middot η]
(ii) Dξ
[partμX
txPξs (y)
](η) = E
[part2μX
txPξs (y ξ ) middot η]
(519)
s isin [t T ] y isin Rd
Furthermore the mixed derivatives partxi(partμXtxPξ (y)) and partμ(partxi
XtxPξ )(y) coin-cide that is
partxi
(partμX
txPξs (y)
) = partμ
(partxi
XtxPξs
)(y) s isin [t T ] P -as
and for
MtxPξs (y z)
= (part2xixj
XtxPξs partμ
(partxi
XtxPξs
)(y) part2
μXtxPξs (y z) partyi
(partμX
txPξs (y)
))
MEAN-FIELD SDES AND ASSOCIATED PDES 863
1 le i le d we have that for all p ge 2 there is some constant Cp isin R such thatfor all t isin [0 T ] x xprime y yprime z zprime and ξ ξ prime isin L2(Ft Rd)
(iii) E[
supsisin[tT ]
∣∣MtxPξs (y z)
∣∣p]le Cp
(iv) E[
supsisin[tT ]
∣∣MtxPξs (y z) minus M
txprimePξ primes
(yprime zprime)∣∣p]
(520)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + ∣∣z minus zprime∣∣p + W2(Pξ Pξ prime)p)
6 Regularity of the value function Given a function isin C21b (Rd times
P2(Rd)) the objective of this section is to study the regularity of the function
V [0 T ] timesRd timesP2(R
d) rarr R
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)]
(61)(t x ξ) isin [0 T ] timesR
d times L2(Ft Rd
)
LEMMA 61 Suppose that isin C11b (Rd timesP2(R
d)) Then under our Hypoth-
esis (H1) V (t middot middot) isin C11b (Rd times P2(R
d)) for all t isin [0 T ] and the derivativespartxV (t xPξ ) = (partxi
V (t xPξ y))1leiled and partμV (t xPξ y) = ((partμV )i(t x
Pξ y))1leiled are of the form
partxiV (t xPξ ) =
dsumj=1
E[(partxj
)(X
txPξ
T PX
tξT
) middot partxiX
txPξ
T j
](62)
(partμV )i(t xPξ y) =dsum
j=1
E[(partxj
)(X
txPξ
T PX
tξT
)(partμX
txPξ
T j
)i (y)
+ E[(partμ)j
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxiX
tyPξ
T j(63)
+ (partμ)j(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T j
)i (y)
]]
Moreover there is some constant C isin R such that for all t t prime isin [0 T ] x xprime yyprime isin R
d and ξ ξ prime isin L2(Ft Rd)
(i)∣∣partxi
V (t xPξ )∣∣ + ∣∣(partμV )i(t xPξ y)
∣∣ le C
(ii)∣∣partxi
V (t xPξ ) minus partxiV
(t xprimePξ prime
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t xprimePξ prime yprime)∣∣
(64)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣ + ∣∣partxiV (t xPξ ) minus partxi
V(t prime xPξ
)∣∣+ ∣∣(partμV )i(t xPξ y) minus (partμV )i
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12
864 BUCKDAHN LI PENG AND RAINER
PROOF In order to simplify the presentation we consider again the case ofdimension d = 1 but without restricting the generality of the argument we use
In accordance with the notation introduced in Section 2 we put V (t x ξ) =V (t xPξ ) and (zϑ) = (zPϑ) (zϑ) isin Rtimes L2(F) Recall also that in the
same sense Xtxξ = XtxPξ Then (XtxξT X
tξT ) = (X
txPξ
T PX
tξT
)
As isin C11b (R times P2(R)) its first-order derivatives are bounded and Lipschitz
continuous Thus standard arguments combined with the results from the preced-ing section show the existence of the Freacutechet derivative Dξ((X
txξT X
tξT )) of the
mapping L2(Ft ) ξ rarr (XtxξT X
tξT ) isin L2(FT ) and for all η isin L2(Ft ) we have
Dξ
(
(X
txξT X
tξT
))(η)
= partx(X
txξT X
tξT
)DξX
txξT (η) + (D)
(X
txξT X
tξT
)(Dξ
[X
tξT
](η)
)= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η](65)
+ E[partμ
(X
txPξ
T PX
tξT
Xt ξT
)Dξ
[X
tξT (η)
]]= partx
(X
txPξ
T PX
tξT
)E
[partμX
txPξ
T (ξ ) middot η]+ E
[partμ
(X
txPξ
T PX
tξT
Xt ξT
) middot (partxX
tξ Pξ
T middot η + E[partμX
tξT (ξ ) middot η])]
(For the notation used here the reader is referred to the previous sections) Withthe argument developed in the study of the first-order derivatives for XtxPξ weconclude that the derivative of (X
txξT P
XtξT
) with respect to the measure in Pξ
is given by
partμ
(
(X
txPξ
T PX
tξT
))(y)
= partx(X
txPξ
T PX
tξT
)partμX
txPξ
T (y)
(66)+ E
[partμ
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
+ partμ(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y)] y isin R
In particular we can deduce from this latter formula and the estimates from thepreceding section that for all p ge 2 there is a constant Cp isin R such that for allx xprime y yprime isin R ξ ξ prime isin L2(Ft )
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y)
∣∣p] le Cp
E[∣∣partμ
(
(X
txPξ
T PX
tξT
))(y) minus partμ
(
(X
txprimePξ primeT P
Xtξ primeT
))(yprime)∣∣p]
(67)
le Cp
(∣∣x minus xprime∣∣p + ∣∣y minus yprime∣∣p + W2(Pξ Pξ prime)p)
MEAN-FIELD SDES AND ASSOCIATED PDES 865
As the expectation E[middot] L2(F) rarr R is a bounded linear operator it followsfrom (65) and (66) that L2(Ft ) ξ rarr V (t x ξ) = V (t xPξ ) =E[(X
txPξ
T PX
tξT
)] is Freacutechet differentiable and for all η isin L2(Ft )
Dξ
[V (t x ξ)
](η) = E
[Dξ
(
(X
txξT X
tξT
))(η)
]= E
[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ ) middot η]]
(68)
= E[E
[partμ
(
(X
txPξ
T PX
tξT
))(ξ )
] middot η]
that is
partμV (t xPξ y) = E[partμ
(
(X
txPξ
T PX
tξT
))(y)
] y isin R(69)
But then from (67) we obtain (64)(i) and (ii) for partμV (t xPξ y)
As concerns the derivative of V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] with respect
to x since z rarr (zPX
tξT
) is a (deterministic) function with a bounded Lipschitz
continuous derivative of first order the computation of partxV (t xPξ ) is standardConcerning the estimates (i) and (ii) for the derivative partxV stated in Lemma 61they are a direct consequence of the assumption on as well as the estimates forthe involved processes studied in the preceding sections
In order to complete the proof it remains still to prove (iii) For this end weobserve that due to Lemma 31 for arbitrarily given (t x) isin [0 T ] times R and ξ isinL2(Ft ) XtxPξ = XtxPξ prime for all ξ prime isin L2(Ft ) with Pξ prime = Pξ Since due to ourassumption L2(F0) is rich enough we can find some ξ prime isin L2(F0) with Pξ prime = Pξ which is independent of the driving Brownian motion B Using the time-shiftedBrownian motion Bt
s = Bt+s minusBt s ge 0 (where we consider the Brownian motionB extended beyond the time horizon T ) we see that XtxPξ prime and Xtξ prime
solve thefollowing SDEs (for simplicity we put b = 0 again)
Xtξ primes+t = ξ prime +
int s
0σ
(X
tξ primer+t PX
tξ primer+t
)dBt
r (610)
XtxPξ primes+t = x +
int s
0σ
(X
txPξ primer+t P
Xtξ primer+t
)dBt
r s isin [0 T minus t](611)
Consequently (XtxPξ primemiddot+t X
tξ primemiddot+t ) and (X0xPξ prime X0ξ prime
) are solutions of the same sys-tem of SDEs only driven by different Brownian motions Bt and B respec-
tively both independent of ξ prime It follows that the laws of (XtxPξ primemiddot+t X
tξ primemiddot+t ) and
(X0xPξ prime X0ξ prime) coincide and hence
V (t xPξ ) = V (t xPξ prime) = E[
(X
txPξ primeT P
Xtξ primeT
)](612)
= E[
(X
0xPξ primeT minust P
X0ξ primeT minust
)]
866 BUCKDAHN LI PENG AND RAINER
Thus for two different initial times t t prime isin [0 T ] using the fact that the derivativesof are bounded that is is Lipschitz over RtimesP2(R) we obtain∣∣V (t xPξ ) minus V
(t prime xPξ
)∣∣le E
[∣∣(X
0xPξ primeT minust P
X0ξ primeT minust
) minus (X
0xPξ primeT minust prime P
X0ξ primeT minust prime
)∣∣](613)
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
le C(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣2])12
But taking into account the boundedness of the coefficient σ of the SDEs forX0xPξ prime and X0ξ prime
we get∣∣V (t xPξ ) minus V(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12(614)
The proof of the remaining estimate (iii) for the derivatives of V is carried outby using the same kind of argument Indeed considering ξ prime isin L2(F0) the sys-tem of equations for NtxPξ prime (y) = (XtxPξ prime partxX
txPξ prime UtxPξ prime (y)) x y isin Rand Ntξ prime
(y) = (Xtξ prime partxX
tξ primeUtξ prime
(y)) y isin R [see (32) (51) (429)] we seeagain that ((
NtxPξ primemiddot+t (y)
)xyisinR
(N
tξ primemiddot+t (y)yisinR
))and ((
N0xPξ prime (y))xyisinR
(N0ξ prime
(y)yisinR))
are equal in lawHence from (69) we deduce
partμV (t xPξ y) = E[partμ
(
(X
txPξ primeT P
Xtξ primeT
))(y)
](615)
= E[partμ
(
(X
0xPξ primeT minust P
X0ξ primeT minust
))(y)
]
Consequently using the Lipschitz continuity and the boundedness of partx R timesP2(R) rarr R and partμ Rtimes P2(R) timesR rarr R as well as the uniform boundednessin Lp (p ge 2) of the first-order derivatives partxX
0xPξ prime partμX0xPξ prime (y) we get from(66) with (0 x ξ prime T minus t) and (0 x ξ prime T minus t prime) instead of (t x ξ T )∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C
(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣ + ∣∣X0yPξ primeT minust minus X
0yPξ primeT minust prime
∣∣ + ∣∣X0ξ primeT minust minus X
0ξ primeT minust prime
∣∣(616)
+ ∣∣partxX0yPξ primeT minust minus partxX
0yPξ primeT minust prime
∣∣ + ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣+ ∣∣partμX
0ξ primePξ primeT minust (y) minus partμX
0ξ primePξ primeT minust prime (y)
∣∣] + W2(PX
0ξ primeT minust
PX
0ξ primeT minust prime
))
MEAN-FIELD SDES AND ASSOCIATED PDES 867
Thus since ξ prime is independent of X0xPξ prime partxX0yPξ prime and partμX0xprimePξ prime (y)∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣le C middot sup
xyisinR(E
[∣∣X0xPξ primeT minust minus X
0xPξ primeT minust prime
∣∣2 + ∣∣partxX0xPξ primeT minust minus partxX
0xPξ primeT minust prime (y)
∣∣2(617)
+ ∣∣partμX0xPξ primeT minust (y) minus partμX
0xPξ primeT minust prime (y)
∣∣2])12
and the uniform boundedness in L2 of the derivatives of X0xPξ prime allows to deducefrom the SDEs for X0xPξ prime partxX
0xPξ prime and partμX0xPξ primeT minust prime (y) [(32) (51) (429)] that∣∣partμV (t xPξ y) minus partμV
(t prime xPξ y
)∣∣ le C∣∣t minus t prime
∣∣12(618)
The proof of the corresponding estimate for partxV (t xPξ ) is similar and henceomitted here
Let us come now to the discussion of the second-order derivatives of our valuefunction V (t xPξ )
LEMMA 62 We suppose that Hypothesis (H2) is satisfied by the coefficientsσ and b and we suppose that isin C
21b (Rd times P2(R
d)) Then for all t isin [0 T ]V (t middot middot) isin C
21b (Rd times P2(R
d)) and the mixed second-order derivatives are sym-metric
partxi
(partμV (t xPξ y)
) = partμ
(partxi
V (t xPξ ))(y)
(t x y) isin [0 T ] timesRd timesR
d ξ isin L2(Ft Rd
)1 le i le d
and for
U(t xPξ y z
) = (part2xixj
V (t xPξ ) partxi
(partμV (t xPξ y)
)
part2μV (t xPξ y z) party
(partμV (t xPξ y)
))
there is some constant C isin R such that for all t t prime isin [0 T ] x xprime y yprime z zprime isin Rd
ξ ξ prime isin L2(Ft Rd)
(i)∣∣U(t xPξ y z)
∣∣ le C
(ii)∣∣U(t xPξ y z) minus U
(t xprimePξ prime yprime zprime)∣∣
(619)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
(iii)∣∣U(t xPξ y z) minus U
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12
PROOF As in the preceding proofs we make our computations for the caseof dimension d = 1 Moreover in our proof we concentrate on the computation
868 BUCKDAHN LI PENG AND RAINER
for the second-order derivative with respect to the measure part2μV (t xPξ y z) and
to its estimates using the preceding lemma on the derivatives of first order thecomputation of the second-order derivatives part2
xV (t xPξ ) partx(partμV (t xPξ )(y))partμ(partxV (t xPξ ))(y) party(partμV (t xPξ )(y)) and their estimates are rather directand left to the interested reader On the other hand a direct computation based on(66) and (69) and using the symmetry of the mixed second-order derivatives of
and of the processes XtxPξ and Xtξ shows that
partx
(partμV (t xPξ y)
) = partμ
(partxV (t xPξ )
)(y)
(t x y) isin [0 T ] timesRtimesR ξ isin L2(Ft )
For the computation of part2μV (t xPξ y z) we use the formula for partμV (t xPξ y)
in Lemma 61 as well as (69) and (66) We observe that
partμV (t xPξ y) = V1(t xPξ y) + V2(t xPξ y) + V3(t xPξ y)(620)
with
V1(t xPξ y) = E[(partx)
(X
txPξ
T PX
tξT
) middot (partμX
txPξ
T
)(y)
]
V2(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
XtyPξ
T
) middot partxXtyPξ
T
]]
V3(t xPξ y) = E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
]]
Let us consider V3(t xPξ y) the discussion for V1(t xPξ y) and V2(t x
Pξ y) is analogous Using the Freacutechet differentiability of the terms involved inthe definition of V3 we obtain for the Freacutechet derivative of
ξ rarr V3(t x ξ y) = V3(t xPξ y)(621)
(t x ξ) isin [0 T ] timesRtimes L2(Ft )
that for all η isin L2(Ft )
DξV3(t x ξ y)(η)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
+ partx(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
txPξ
T
](η)(622)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot Dξ
[X
tξ
T
](η)
] middot (partμX
tξ Pξ
T
)(y)
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot Dξ
[X
tξT
](η)
]]
MEAN-FIELD SDES AND ASSOCIATED PDES 869
(recall the notation introduced in Section 4) On the other hand we know alreadythat
(i) Dξ
[X
txPξ
T
](η) = E
[partμX
txPξ
T (ξ ) middot η]
(ii) Dξ
[X
tξ
T
](η) = partxX
tξPξ
T middot η + E[partμX
tξPξ
T (ξ ) middot η]
(iii) Dξ
[X
tξT
](η) = partxX
tξ Pξ
T middot η + E[partμX
tξ Pξ
T (ξ ) middot η](623)
(iv) Dξ
[(partμX
tξ Pξ
T
)(y)
](η)
= partx
(partμX
tξ Pξ
T (y)) middot η + E
[(part2μX
tξ Pξ
T
)(y ξ ) middot η]
Consequently we have
DξV3(t x ξ y)(η)
= E[E
[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tξ Pξ
T (y))
+ (part2μX
tξ Pξ
T
)(y ξ )
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y) middot partμX
txPξ
T (ξ )
(624)+ E
[(part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξPξ
T (ξ ))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tξ Pξ
T + partμXtξ Pξ
T (ξ ))]] middot η]
Therefore
partμV3(t xPξ y z)
= E[E
[(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot (partx
(partμX
tzPξ
T (y)) + (
part2μX
tξ Pξ
T
)(y z)
)+ partx(partμ)
(X
txPξ
T PX
tξT
Xt ξT
) middot partμXtξ Pξ
T (y) middot partμXtxPξ
T (z)
+ E[(
part2μ
)(X
txPξ
T PX
tξT
Xt ξT X
tξ
T
) middot partμXtξ Pξ
T (y)(625)
times (partxX
tzPξ
T + partμXtξPξ
T (z))]
+ party(partμ)(X
txPξ
T PX
tξT
Xt ξT
) middot (partμX
tξ Pξ
T
)(y)
times (partxX
tzPξ
T + partμXtξ Pξ
T (z))]]
870 BUCKDAHN LI PENG AND RAINER
t isin [0 T ] x y z isin R ξ isin L2(Ft ) satisfies the relation
DV3(t x ξ y)(η) = E[partμV3(t xPξ y ξ ) middot η]
(626)
that is the function partμV3(t xPξ y z) is the derivative of V3(t xPξ y) with re-spect to the measure at Pξ Moreover the above expression for partμV3(t xPξ y z)
combined with the estimates for the process XtxPξ and those of its first- andsecond-order derivatives studied in the preceding sections allows to obtain after adirect computation∣∣partμV3(t xPξ y z) minus partμV3
(t xprimeP prime
ξ yprime zprime)∣∣
(627)le C
(∣∣x minus xprime∣∣ + ∣∣y minus yprime∣∣ + ∣∣z minus zprime∣∣ + W2(Pξ Pξ prime))
for all t isin [0 T ] x xprime y yprime z zprime isin R and ξ ξ prime isin L2(Ft ) Furthermore extendingin a direct way the corresponding argument for the estimate of the difference for thefirst-order derivatives of V (t xPξ ) at different time points [see (616)] we deducefrom the explicit expression for partμV3(t xPξ y z) that for some real C isin R∣∣partμV3(t xPξ y z) minus partμV3
(t prime xPξ y z
)∣∣ le C∣∣t minus t prime
∣∣12(628)
for all t t prime isin [0 T ] x y z isin R and ξ isin L2(Ft )In the same manner as we obtained the wished results for partμV3(t xPξ y z)
we can investigate partμV1(t xPξ y z) and partμV2(t xPξ y z) This yields thewished results for part2
μV (t xPξ y z) The proof is complete
7 Itocirc formula and PDE associated with mean-field SDE Let us begin withestablishing the Itocirc formula which will be applied after for the study of the PDEassociated with our mean-field SDE
THEOREM 71 Let F [0 T ] times Rd times P(Rd) rarr R be such that F(t middot middot) isin
C21b (Rd times P(Rd)) for all t isin [0 T ] F(middot xμ) isin C1([0 T ]) for all (xμ) isin
Rd times P2(R
d) and all derivatives with respect to t of first order and with re-spect to (xμ) of first and of second order are uniformly bounded over [0 T ] timesR
d times P(Rd) (for short F isin C1(21)b ([0 T ] times R
d times P2(Rd))) Then under Hy-
pothesis (H2) for all 0 le t le s le T x isin Rd ξ isin L2(Ft Rd) the following Itocirc
formula is satisfied
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
=int s
t
(partrF
(rX
txPξr P
Xtξr
) +dsum
i=1
partxiF
(rX
txPξr P
Xtξr
)bi
(X
txPξr P
Xtξr
)
+ 1
2
dsumijk=1
part2xixj
F(rX
txPξr P
Xtξr
)(σikσjk)
(X
txPξr P
Xtξr
)(71)
MEAN-FIELD SDES AND ASSOCIATED PDES 871
+ E
[dsum
i=1
(partμF )i(rX
txPξr P
Xtξr
Xt ξr
)bi
(Xtξ
r PX
tξr
)
+ 1
2
dsumijk=1
partyi(partμF )j
(rX
txPξr P
Xtξr
Xt ξr
)(σikσjk)
(Xtξ
r PX
tξr
)])dr
+int s
t
dsumij=1
partxiF
(rX
txPξr P
Xtξr
)σij
(X
txPξr P
Xtξr
)dBj
r s isin [t T ]
PROOF As already before in other proofs let us again restrict ourselves todimension d = 1 The general case is got by a straight-forward extension
Step 1 Let us begin with considering a function f isin C21b (P2(R)) let ξ isin
L2(Ft ) and 0 le t lt sz le T We put tni = t + i(s minus t)2minusn0 le i le 2n n ge 1Due to Lemma 21 we have
f (PX
tξs
) minus f (Pξ )
=2nminus1sumi=0
(f (P
Xtξ
tni+1
) minus f (PX
tξ
tni
))
=2nminus1sumi=0
(E
[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)](72)
+ 1
2E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]+ 1
2E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2] + Rtni(P
Xtξ
tni+1
PX
tξ
tni
)
)(for the notation we refer to the preceding sections) where for some C isin R+depending only on the Lipschitz constants of part2
μf and partypartμf
∣∣Rtni(P
Xtξ
tni+1
PX
tξ
tni
)∣∣ le CE
[∣∣Xtξ
tni+1minus X
tξ
tni
∣∣3]le C
(E
[(int tni+1
tni
∣∣b(X
txPξr P
Xtξr
)∣∣dr
)3](73)
+ E
[(int tni+1
tni
∣∣σ (X
txPξr P
Xtξr
)∣∣2 dr
)32])le C
(tni+1 minus tni
)32 0 le i le 2n minus 1
872 BUCKDAHN LI PENG AND RAINER
Thus taking into account the relations (recall that B and B are independent Brow-nian motions)
(i) E[partμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)]
= E
[partμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
b(Xtξ
r PX
tξr
)dr
]
(ii) E[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)]]= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
σ(X
tξ
r PX
tξr
)dBr +
int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
= E
[E
[part2μf
(P
Xtξ
tni
Xt ξ
tniX
tξ
tni
) middot(int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)
times(int tni+1
tni
b(X
tξ
r PX
tξr
)dr
)]]
(iii) E[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(X
tξ
tni+1minus X
tξ
tni
)2]= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
)(int tni+1
tni
σ(Xtξ
r PX
tξr
)dBr
+int tni+1
tni
b(Xtξ
r PX
tξr
)dr
)2]
= E
[partypartμf
(P
Xtξ
tni
Xt ξ
tni
) middotint tni+1
tni
∣∣σ (Xtξ
r PX
tξr
)∣∣2 dr
]+ Qtni
with |Qtni| le C(tni+1 minus tni )32 0 le i le 2n minus 1 as well as the continuity of r rarr
(XtxPξr P
Xtξr
) isin L2(FRtimesP2(R)) we get from the above sum over the second-
MEAN-FIELD SDES AND ASSOCIATED PDES 873
order Taylor expansion as n rarr +infin
f (PX
tξs
) minus f (Pξ )
=int s
tE
[b(Xtξ
r PX
tξr
)partμf
(P
Xtξr
Xt ξr
)]dr(74)
+ 1
2
int s
tE
[σ 2(
Xtξr P
Xtξr
)(partypartμf )
(P
Xtξr
Xt ξr
)]dr s isin [t T ]
From this latter relation we see that for fixed (t ξ) the function s rarr f (PX
tξs
) s isin[t T ] is continuously differentiable in s and twice continuously differentiable andin particular
partsf (PX
tξs
) = E[b(Xtξ
s PX
tξs
)partμf
(P
Xtξs
Xt ξs
)(75)
+ 12σ 2(
Xtξs P
Xtξs
)(partypartμf )
(P
Xtξs
Xt ξs
)] s isin [t T ]
Step 2 From the preceding step we can derive that for F isin C1(21)b ([0 T ] times
RtimesP2(R)) also (s x) = F(s xPX
tξs
) is continuously differentiable
parts(s x) = (partsF )(s xPX
tξs
) + E[b(Xtξ
s PX
tξs
)partμF
(s xP
Xtξs
Xt ξs
)+ 1
2σ 2(Xtξ
s PX
tξs
)(partypartμF )
(s xP
Xtξs
Xt ξs
)]
and twice continuously differentiable with respect to x Hence we can apply to
(sXtxPξs )(= F(sX
txPξs P
Xtξs
)) the classical Itocirc formula This yields
F(sX
txPξs P
Xtξs
) minus F(t xPξ )
= (sX
txPξs
) minus (t x)
=int s
t
(partr
(rX
txPξr
) + b(X
txPξr P
Xtξr
)partx
(rX
txPξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2x
(rX
txPξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(rX
txPξr
)dBr
=int s
t
(partrF
(rX
txPξr P
Xtξr
)(76)
+ E
[b(Xtξ
r PX
tξr
)partμF
(rX
txPξr P
Xtξr
Xt ξr
)+ 1
2σ 2(
Xtξr P
Xtξr
)(partypartμF )
(rX
txPξr P
Xtξr
Xt ξr
)]
874 BUCKDAHN LI PENG AND RAINER
+ b(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)+ 1
2σ 2(
XtxPξr P
Xtξr
)part2xF
(rX
txPξr P
Xtξr
))dr
+int s
tσ
(X
txPξr P
Xtξr
)partx
(X
txPξr P
Xtξr
)dBr s isin [t T ]
The proof is complete
The above Itocirc formula allows now to show that our value function V (t xPξ ) iscontinuously differentiable with respect to t with a derivative parttV bounded over[0 T ] timesR
d timesP2(Rd)
LEMMA 71 Assume that isin C21(Rd times P2(Rd)) Then under Hypothe-
sis (H2) V isin C1(21)([0 T ] times Rd times P2(R
d)) and its derivative parttV (t xPξ )
with respect to t verifies for some constant C isin R
(i)∣∣parttV (t xPξ )
∣∣ le C
(ii)∣∣parttV (t xPξ ) minus parttV
(t xprimePξ prime
)∣∣ le C(∣∣x minus xprime∣∣ + W2(Pξ Pξ prime)
)(77)
(iii)∣∣parttV (t xPξ ) minus parttV
(t prime xPξ
)∣∣ le C∣∣t minus t prime
∣∣12
for all t t prime isin [0 T ] x xprime isin Rd ξ ξ prime isin L2(Ft Rd)
PROOF Recall that for t isin [0 T ] x isin Rd and ξ [which can be supposed
without loss of generality to belong to L2(F0) see our previous discussion in theproof of Lemma 61] we have
V (t xPξ ) = E[
(X
txPξ
T PX
tξT
)] = E[
(X
0xPξ
T minust PX
0ξT minust
)](78)
Hence taking the expectation over the Itocirc formula in the preceding theorem forF(s xPξ ) = (xPξ ) s = T minus t and initial time 0 we get with for simplicityd = 1 again
V (t xPξ ) minus V (T xPξ )
=int T minust
0E
[(partx
(X
0xPξr P
X0ξr
)b(X
0xPξr P
X0ξr
)+ 1
2part2x
(X
0xPξr P
X0ξr
)σ 2(
X0xPξr P
X0ξr
)(79)
+ E
[(partμ)
(X
0xPξr P
X0ξs
X0ξr
)b(X0ξ
r PX
0ξr
)+ 1
2party(partμ)
(X
0xPξr P
X0ξr
X0ξr
)σ 2(
X0ξr P
X0ξr
)])]dr
MEAN-FIELD SDES AND ASSOCIATED PDES 875
Then it is evident that V (t xPξ ) is continuously differentiable with respect to t
parttV (t xPξ ) = minusE[partx
(X
0xPξ
T minust PX
0ξT minust
)b(X
0xPξ
T minust PX
0ξT minust
)+ 1
2part2x
(X
0xPξ
T minust PX
0ξT minust
)σ 2(
X0xPξ
T minust PX
0ξT minust
)(710)
+ E[(partμ)
(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)b(X
0ξT minust PX
0ξT minust
)+ 1
2party(partμ)(X
0xPξ
T minust PX
0ξT minust
X0ξT minust
)σ 2(
X0ξT minust PX
0ξT minust
)]]
Moreover using this latter formula we can now prove in analogy to the otherderivatives of V that parttV satisfies the estimates stated in this lemma The proof iscomplete
Now we are able to establish and to prove our main result
THEOREM 72 We suppose that isin C21b (Rd times P2(R
d)) Then under
Hypothesis (H2) the function V (t xPξ ) = E[(XtxPξ
T PX
tξT
)] (t x ξ) isin[0 T ]timesR
d timesL2(Ft Rd) is the unique solution in C1(21)([0 T ]timesRd timesP2(R
d))
of the PDE
0 = parttV (t xPξ ) +dsum
i=1
partxiV (t xPξ )bi(xPξ )
+ 1
2
dsumijk=1
part2xixj
V (t xPξ )(σikσjk)(xPξ )
+ E
[dsum
i=1
(partμV )i(t xPξ ξ)bi(ξPξ )
(711)
+ 1
2
dsumijk=1
partyi(partμV )j (t xPξ ξ)(σikσjk)(ξPξ )
]
(t x ξ) isin [0 T ] timesRd times L2(
Ft Rd)
V (T xPξ ) = (xPξ ) (x ξ) isin Rd times L2(
FRd)
PROOF As before we restrict ourselves in this proof to the one-dimensionalcase d = 1 Recalling the flow property
(X
sXtxPξs P
Xtξs
r XsXtξs
r
) = (X
txPξr Xtξ
r
)
(712)0 le t le s le r le T x isin R ξ isin L2(Ft )
876 BUCKDAHN LI PENG AND RAINER
of our dynamics as well as
V (s yPϑ) = E[
(X
syPϑ
T PX
sϑT
)] = E[
(X
syPϑ
T PX
sϑT
)|Fs
](713)
s isin [0 T ] y isinR ϑ isin L2(Fs) we deduce that
V(sX
txPξs P
Xtξs
) = E[
(X
syPϑ
T PX
sϑT
)|Fs
]|(yϑ)=(X
txPξs X
tξs )
= E[
(X
sXtxPξs P
Xtξs
T PX
sXtξs
T
)|Fs
](714)
= E[
(X
txPξ
T PX
tξT
)|Fs
] s isin [t T ]
that is V (sXtxPξs P
Xtξs
) s isin [t T ] is a martingale On the other hand since
due to Lemma 71 the function V isin C1(21)b ([0 T ] times R
d times P2(Rd)) satisfies the
regularity assumptions for the Itocirc formula we know that
V(sX
txPξs P
Xtξs
) minus V (t xPξ )
=int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(715)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr
+int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
Consequently
V(sX
txPξs P
Xtξs
) minus V (t xPξ )(716)
=int s
tpartxV
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr s isin [t T ]
and
0 =int s
t
(partrV
(rX
txPξr P
Xtξr
) + partxV(rX
txPξr P
Xtξr
)b(X
txPξr P
Xtξr
)+ 1
2part2xV
(rX
txPξr P
Xtξr
)σ 2(
XtxPξr P
Xtξr
)(717)
+ E
[partμV
(rX
txPξr P
Xtξr
Xt ξr
)b(Xtξ
r PX
tξr
)+ 1
2party(partμV )
(rX
txPξr P
Xtξr
Xt ξr
)σ 2(
Xtξr P
Xtξr
)])dr s isin [t T ]
MEAN-FIELD SDES AND ASSOCIATED PDES 877
from where we obtain easily the wished PDEThus it only still remains to prove the uniqueness of the solution of the PDE in
the class C1(21)b ([0 T ]timesR
d timesP2(Rd)) Let U isin C
1(21)b ([0 T ]timesR
d timesP2(Rd))
be a solution of PDE (711) Then from the Itocirc formula we have that
U(sX
txPξs P
Xtξs
) minus U(t xPξ )
(718)=
int s
tpartxU
(rX
txPξr P
Xtξr
)σ
(X
txPξr P
Xtξr
)dBr
s isin [t T ] is a martingale Thus for all t isin [0 T ] x isin R and ξ isin L2(Ft )
U(t xPξ ) = E[U
(T X
txPξ
T PX
tξT
)|Ft
](719)
= E[
(X
txPξ
T PX
tξT
)] = V (t xPξ )
This proves that the functions U and V coincide that is the solution is unique inC
1(21)b ([0 T ] timesR
d timesP2(Rd)) The proof is complete
Acknowledgments The authors would like to thank the anonymous Asso-ciate Editor and the anonymous referees for their very helpful comments and sug-gestions from which the manuscript greatly has benefited
REFERENCES
[1] BENACHOUR S ROYNETTE B TALAY D and VALLOIS P (1998) Nonlinear self-stabilizing processes I Existence invariant probability propagation of chaos StochasticProcess Appl 75 173ndash201 MR1632193
[2] BENSOUSSAN A FREHSE J and YAM S C P (2015) Master equation in mean-fieldtheorie Preprint Available at arXiv14044150v1
[3] BOSSY M and TALAY D (1997) A stochastic particle method for the McKeanndashVlasov andthe Burgers equation Math Comp 66 157ndash192 MR1370849
[4] BUCKDAHN R DJEHICHE B LI J and PENG S (2009) Mean-field backward stochasticdifferential equations A limit approach Ann Probab 37 1524ndash1565 MR2546754
[5] BUCKDAHN R LI J and PENG S (2009) Mean-field backward stochastic differential equa-tions and related partial differential equations Stochastic Process Appl 119 3133ndash3154MR2568268
[6] CARDALIAGUET P (2013) Notes on mean field games (from P L Lionsrsquo lectures at Collegravegede France) Available at httpswwwceremadedauphinefr~cardalia
[7] CARDALIAGUET P (2013) Weak solutions for first order mean field games with local cou-pling Preprint Available at httphalarchives-ouvertesfrhal-00827957
[8] CARMONA R and DELARUE F (2014) The master equation for large population equilibri-ums In Stochastic Analysis and Applications 2014 Springer Proc Math Stat 100 77ndash128 Springer Cham MR3332710
[9] CARMONA R and DELARUE F (2015) Forward-backward stochastic differential equationsand controlled McKeanndashVlasov dynamics Ann Probab 43 2647ndash2700 MR3395471
[10] CHAN T (1994) Dynamics of the McKeanndashVlasov equation Ann Probab 22 431ndash441MR1258884
878 BUCKDAHN LI PENG AND RAINER
[11] DAI PRA P and DEN HOLLANDER F (1996) McKeanndashVlasov limit for interacting randomprocesses in random media J Stat Phys 84 735ndash772 MR1400186
[12] DAWSON D A and GAumlRTNER J (1987) Large deviations from the McKeanndashVlasov limitfor weakly interacting diffusions Stochastics 20 247ndash308 MR0885876
[13] KAC M (1956) Foundations of kinetic theory In Proceedings of the Third Berkeley Sym-posium on Mathematical Statistics and Probability 1954ndash1955 Vol III 171ndash197 UnivCalifornia Press Berkeley MR0084985
[14] KAC M (1959) Probability and Related Topics in Physical Sciences Interscience PublishersLondon MR0102849
[15] KOTELENEZ P (1995) A class of quasilinear stochastic partial differential equations ofMcKeanndashVlasov type with mass conservation Probab Theory Related Fields 102 159ndash188 MR1337250
[16] LASRY J-M and LIONS P-L (2007) Mean field games Jpn J Math 2 229ndash260MR2295621
[17] LIONS P L (2013) Cours au Collegravege de France Theacuteorie des jeu agrave champs moyens Availableat httpwwwcollege-de-francefrdefaultENallequ[1]deraudiovideojsp
[18] MEacuteLEacuteARD S (1996) Asymptotic behaviour of some interacting particle systems McKeanndashVlasov and Boltzmann models In Probabilistic Models for Nonlinear Partial Differen-tial Equations (Montecatini Terme 1995) Lecture Notes in Math 1627 42ndash95 SpringerBerlin MR1431299
[19] OVERBECK L (1995) Superprocesses and McKeanndashVlasov equations with creation of massPreprint
[20] SZNITMAN A-S (1984) Nonlinear reflecting diffusion process and the propagation of chaosand fluctuations associated J Funct Anal 56 311ndash336 MR0743844
[21] SZNITMAN A-S (1991) Topics in propagation of chaos In Eacutecole DrsquoEacuteteacute de Probabiliteacutesde Saint-Flour XIXmdash1989 Lecture Notes in Math 1464 165ndash251 Springer BerlinMR1108185
R BUCKDAHN
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
AND
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL rainerbuckdahnuniv-brestfr
J LI
SCHOOL OF MATHEMATICS AND STATISTICS
SHANDONG UNIVERSITY WEIHAI
WEIHAI SHANDONG PROVINCE 264209P R CHINA
E-MAIL juanlisdueducn
S PENG
SCHOOL OF MATHEMATICS
SHANDONG UNIVERSITY
JINAN SHANDONG PROVINCE 250100P R CHINA
E-MAIL pengsdueducn
C RAINER
LABORATOIRE DE MATHEacuteMATIQUES
CNRS-UMR 6205UNIVERSITEacute DE BRETAGNE OCCIDENTALE
6 AVENUE VICTOR-LE-GORGEU
BP 809 29285 BREST CEDEX
FRANCE
E-MAIL catherineraineruniv-brestfr
- Introduction
- Preliminaries
- The mean-field stochastic differential equation
- First-order derivatives of XtxPxi
- Second-order derivatives of XtxPxi
- Regularity of the value function
- Itocirc formula and PDE associated with mean-field SDE
- Acknowledgments
- References
- Authors Addresses
-