existence and necessary conditions in the control of … · existence and necessary conditions in...
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SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
EXISTENCE AND NECESSARY CONDITIONSIN THE CONTROL OF FBSDE
Brahim Mezerdi
Laboratory of Applied Mathematics
University of Biskra, Algeria
Workshop on Finance and Insurance
Jena, March 16-20, 2008
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
SUMMARY
Joint work with K. Bahlali (Univ. Toulon France) and B. Gherbal (Univ.Biskra)
I Formulation of the problem
I The Relaxed Problem
I Existence result
I Maximum principle
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Formulation of the problem
The systems we wish to control are driven by the following d-dimensionalFBSDE
dXt = b (t,Xt ,Ut) dt + σ (t,Xt) dWt ,X (0) = x ,−dYt = f (t,Xt ,Yt ,Ut) dt − ZtdMX
t ,Y (T ) = g (XT ) .
(1.1)
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Formulation of the problem
b, σ, f and g are given, MX is the martingale part of the diffusionprocess X , (Wt , t ≥ 0) is a standard Brownian motion, defined on(Ω,F , (Ft)t≥0 ,P
),. The control variable ut , called strict control, is an
Ft adapted process with values in some compact metric space A.The expected cost on the time interval [0,T ] is of the form
J (U) = E
[l (Y0) +
∫ T
0
h (t,Xt ,Yt ,Ut) dt
]. (1.2)
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Formulation of the problem
Let (Wt , t ≥ 0) be a d-dimensional Brownian motion defined on some
filtered probability space(Ω,F , (Ft)t≥0 ,P
).
(A1) Assume that
b : [0,T ]× Rd × A → Rd ,
σ : [0,T ]× Rd → Rd×d ,
f : [0,T ]× Rd × Rd × A → Rk
g : [0,T ]× Rd → Rk
are bounded, measurable and continuous.We assume also that f is uniformly Lipschitz in (x , y) , uniformly withrespect to u.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Formulation of the problem
We can reformulate the above control problem as an equivalentmartingale problem. This simplifies taking limits. Let L be theinfinitesimal generator, associated with (2.1), acting on functions ϕ inC 2
b
(Rd ; R
)defined by
Lϕ (t, x , u) :=1
2
∑i,j
(aij
∂2ϕ
∂xi∂xj
)(t, x) +
∑i
(bi∂ϕ
∂xi
)(t, x , u) , (2.2)
where aij (t, x , u) denotes the generic term of the symmetric matrixσσ∗ (t, x , u).
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Formulation of the problem
Definition 2.1. A strict control is a collectionU = (Ω,F , Ftt ,P,Zt ,Xt , x ,Yt ,Ut) such that(1) x ∈ Rd is the initial data,
(2)(Ω,F , Ftt10 ,P
)is a filtered probability space satisfying the
usual conditions,(3) Ut is a A-valued process, Ft-progressively measurable,
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Formulation of the problem
(4) (Xt)t is an Rd -valued, Ft-adapted, with continuous paths, such thatX (0) = x and satisfies for each ϕ ∈ C 2
b
(Rd ,R
),
ϕ (Xt)− ϕ (x)−∫ t
0
Lϕ (s,Xs ,Ut) ds is a P-martingale, (2.3)
where L is the infinitesimal generator defined by (2.2),(5) (Yt)t is Rd -valued, Ft-adapted, such that
Yt = g (XT ) +
∫ T
t
f (s,Xs ,Ys ,Ut) ds −∫ T
t
ZsdMXs , 0 ≤ t ≤ T , (2.4)
(6) MXt is the martingale part of the diffusion process X and f (., y , u) is
F t-progressively measurable.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Formulation of the problem
We denote by U the set of strict controls.The cost corresponding to a control U is defined by
J (U) = E
(l (Y0) +
∫ T
0
h (t,Xt ,Yt ,Ut) dt
)(2.5)
whereA3)
h : [0,T ]× Rd × Rd × A → Rk
l : Rd → R
are bounded and continuous functions.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The relaxed control problem
The idea of relaxed control is to replace the A-valued process (Ut) withP (A)-valued process (qt), where P (A) is the space of probabilitymeasures equipped with the topology of weak convergence. Let V be theset of probability measures on [0,T ]× A whose projections on [0,T ]coincide with the Lebesgue measure dt. Equipped with the topology ofstable convergence of measures, V is a compact metrizable space, seeJacod & Memin.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The relaxed control problem
Definition 3.1. A relaxed control is a collectionq :=
(Ω, F , F tt10 ,P,Zt ,Xt , x , ,Yt , qt
)such that
(1) x ∈ Rd is the initial data,
(2)(Ω,F , Ftt10 ,P
)is a filtered probability space satisfying the
usual conditions,(3) (qt)t is a P (A)-valued process, Ft-progressively measurable suchthat for each t, 1(0,t] · q is Ft-measurable.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The relaxed control problem
(4) (Xt)t is an Rd -valued, adapted, with continuous paths, satisfyingX (0) = x and for each ϕ ∈ C 2
b
(Rd ,R
),
ϕ (Xt)− ϕ (x)−∫ t
0
∫A
Lϕ (s,Xs , a) qs (da) ds is a P-martingale. (3.1)
(5) (Yt)t is Rd -valued, Ft-adapted, such that
Yt = g (XT ) +
∫ T
t
∫A
f (s,Xs ,Ys , a) qs (da) ds −∫ T
t
ZsdMXs . (3.2)
(6) MXt is the martingale part of the forward component X .
We denote by R the collection of all relaxed controls.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The relaxed control problem
The cost function associated to a relaxed control q is defined by
J (q) = E
(l (Y0) +
∫ T
0
∫A
h (t,Xt ,Yt , a) qt (da) dt
)(3.3)
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
The main result of this section is given by the following.
Theorem 4.1. Under conditions A1 and A2, there exists a relaxedcontrol q ∈ R such that
J (q) = infµ∈R
J (µ) . (4.0)
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
Jakubowsky S-TopologyThe S-topology has been introduced by Jakubowski, as a topologydefined on the Skorokhod space of cadlag functions D
([0,T ] ; Rk
). This
topology is weaker than the Skorokhod topology and the tightnesscriteria are easier to establish. These criteria are the same as the oneused in Meyer & Zheng topology.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
A.1)Write xn →S x0, if for every ε > 0 one can find functions νn,ε ofbound ed variation in [0,T ] , which are ε-uniformly close to xn’s andweakly-* convergent
supt∈[0,T ]
|xn (t)− νn,ε (t)| ≤ ε, n = 0, 1, 2, ...
vn,ε →wν0,ε, as n →∞
A.2) K ⊂ D ([0,T ]) is relatively S-compact if and only if, the followingconditions hold
supK
supt∈[0,T ]
|x (t)| ≤ CK <∞
andsupK
Nη (x) ≤ Cη <∞ η > 0,
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
We recall (see Jakubowski ) that for a familly (X n)n of quasi-martingales
on the probability space((
Ω, Ft0≤t≤T ,P))
, the following condition
insures the tightness of the familly (X n)n on the space D([0,T ] ; Rk
)endowed with the S-topology
supn
(sup
0≤t≤TE |X n
t |+ CV 0t (X n)
)<∞,
CV 0t (X ) = supE
(∑i
∣∣E (Xti+1 − Xti ) Fnti
∣∣) ,
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
To prove the existence theorem, we need some results on the tightness ofprocesses under consideration.
Let (qn)n≥0 be a minimizing sequence, that is limn→∞
J (qn) = infµ∈R
J (µ) .
Let (X n,Y n,Z n) be the solution of (3.1) and (3.2) corresponding to qn.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
Lemma 4.4. The family of relaxed controls (qn)n≥0 is tight in V.
Proposition 4.5. Let X nt be such that
X nt =
∫ t
0
∫A
b (s,X ns , a) qn
s (da) ds +
∫ t
0
σ (s,X ns ) dWs . (4.6)
Then, X n is tight in the space C([0,T ] ,Rk
)endowed with the
topology of the uniform convergence.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
Proposition 4.2. There exists a positive constant C such that
supn
E(
sup0≤t≤T
|Y nt |
2 +
∫ T
t
|Z ns σ(X n
s )|2ds< C (4.1)
Proposition 4.3. The sequence (Y n,∫ ·0Z n
s dMX n
s ) is tight on the space
D([0,T ] ; Rk
)× D
([0,T ] ; Rk
)endowed with the S-topology.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
Sketch of proofLet (qn)n≥0 be the minimizing sequence
ϕ (X nt )− ϕ (x)−
∫ t
0
∫A
Lϕ (s,X ns , a) qn
s (da) ds is a P-martingale
Y nt = g (X n
T ) +
∫ T
t
∫A
f (s,X ns ,Y
ns , a) qn
s (da) ds + Mnt −Mn
T , (4.7)
where Mnt :=
∫ t
0Z n
s dMX n
s , and
limn→∞
J (qn) = limn→∞
E
[l (Y n
0 ) +
∫ T
0
∫A
h (t,X nt ,Y
nt , a) qn
t (da) dt
]= λ.
(4.8)
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
From tightness results, it follows that the sequence of processes
γn = (qn,X n,Y n,Mn) (4.9)
is tight on the space
Γ = V×C([0,T ] ,Rk
)×[D([0,T ] ; Rk
)]2(4.10)
equipped with the product topology of the uniform convergence on thefirst factor, the topology of stable convergence of measures on thesecond factor and the S-topology on the third factor.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
By Jakubowski’s theorem, there exists a probability space(Ω, F , P
), a
sequence γn =(X n, qn, Y n, Mn
)and γ =
(X , q, Y , M
)defined on this
space such that(i) for each n ∈ N, law(γn) = law(γn),(ii) there exists a subsequence (γnk ) of (γn) , still denoted (γn) ,
which converges to γ,P-a.s. on the space Γ,(iii) Y n
t converges to Yt , dt × P − a.s.,
(iv) sup0≤t≤T
∣∣∣X nt − Xt
∣∣∣→ 0, as n →∞, P − a.s.
To conclude we pass to the limit in the martingale problem for theforward component and in the BSDE.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
Existence
We consider the following assumption:
(A2) (Roxin’s condition): For every (t, x , y) ∈ [0,T ]× Rn × Rm, the set
(b, f , h) (t, x , y ,A)
:= bi (t, x , u) , fj (t, x , y , u) , h (t, x , y , u) u ∈ A, i = 1, ..., n, j = 1, ...,m ,
is convex and closed in Rn+m+1.
Corollary Suppose that assumptions A1, A2, A3 hold, then the relaxedoptimal control qt has the form of a Dirac measure charging a strictcontrol Ut (i.e; dtqt (da) = dtδUt
(da)).
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
Throughout this section, the following assumptions will be in force.(A4) b, σ, f , g , h and l are continuous in [0,T ]× Rn × Rm × Rk and
continuously differentiable with respect to x , y .
(A5) The derivatives of b, σ, f , g , h and l with respect to x , y areuniformly bounded,
|ρ(t, x , y , u)| ≤ c for ρ = bx , σx , fx , fy , gx , hx , hy
and|ly (t, x , y , u)| ≤ c (1 + |y |)
(A6) there exists a positive constant c such that for every(t, x , x ′, y , y ′, u)
|bx (t, x , u)− bx (t, x ′, u)|+ |σx (t, x)− σx (t, x ′)| ≤ c |x − x ′|
and|ρ (t, x , y , u)− ρ (t, x ′, y ′, u)| ≤ c (|x − x ′|+ |y − y ′|) ,
for ρ = fx , fy , hx , hy .
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
The set U of strict controls is embedded into the set R of relaxedcontrols, by the mapping
ψ : u ∈ U 7→ ψ (u) (dt, da) = dtδu(t) (da) ∈ R, (5.1)
Lemma 5.2. (chattering lemma ). Let (µt) be a predictable processwith values in the space of probability measures on A. Then there existsa sequence of predictable processes (Un
t ) with values in A, such that thesequence of random measures
(δUn
t(da) dt
)converges weakly to
µt (da) dt,P-a.s.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
Denote by (X nt ,Y
nt ,Z
nt ) the solution of FBSDE which which can be
written in the relaxed form asdX n
t =∫A
b (t,X nt , a)µ
nt (da) dt + σ (t,X n
t ) dWt , X n0 = x ,
−dY nt =
∫A
f (t,X nt ,Y
nt , a)µ
nt (da) dt − Z n
t dMX n
t , Y nT = g (X n
T ) .(5.5)
Where µnt (da) = δUn(t) (da) and MX n
t is the martingale part of thediffusion X n.
Theorem 5.3. Let (Xt ,Yt ,Zt) and (X nt ,Y
nt ,Z
nt ) be the solutions of
((5.2),(5.3)) and (5.5) associated with µ and Un, respectively. Then itfollows that
limn→+∞
E
(sup
0≤t≤T|X n
t − Xt |2)
= 0, (5.6)
and
limn→+∞
E
(|Y n
t − Yt |2 +
∫ T
0
|Z nt σ (t,X n
t )− Ztσ (t,Xt)|2 dt
)= 0. (5.7)
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
Proposition 5.4. Let J (Un) and J (µ) be the expected costscorresponding respectively to Un and µ, where Un and µ are defined asin the last theorem. Then there exist a subsequence (Unk ) of (Un) , stilldenoted by (Un) such that J (Un) converges to J (µ) .
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
We know from the last section that an optimal relaxed control exists andthat there is a sequence (un) ∈ U of strict controls such that
dtqnt (da) = dtδun
t(da) →
n→+∞dtqt (da) P-a.s in V,
According to the optimality of q, there exists a sequence (εn) of positivereal numbers with lim
n→+∞εn = 0 such that
J (un) = J (qn) ≤ J (q) + εn, (5.8)
where qn = dtδunt(da) .
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
To derive necessary conditions for near optimality, we use Ekeland’svariational principle along with an appropriate choice of a metric on thespace U of admissible controls.
Define a metric d on the space U by
d (u, v) := P ⊗ dt
(ω, t) ∈ Ω× [0,T ] ; u (ω, t) 6= v (ω, t), (5.9)
where P ⊗ dt is the product measure of P and the Lebesgue measure.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
Let us summarize some of the properties satisfied by d
Lemma i) (U, d) is a complete metric space.ii) For any u, v ∈ U along with the corresponding trajectories(X ,Y ,Z ) , (X ′,Y ′,Z ′), it hold that
E
(sup
t∈[0,T ]
|Xt − X ′t |
2
)≤ C1 (d (u, v))
12 (5.10)
and
E
(sup
t∈[0,T ]
|Yt − Y ′t |
2+
∫ T
0
|Ztσ (t,Xt)− Z ′tσ (t,X ′t )|
2dt
)≤ C1 (d (u, v))
12
(5.11)iii) The cost functional J : (U, d) → R is continuous. More precisely ifu, v are two elements in U then
|J (u)− J (v)| ≤ C (d (u, v))12 . (5.12)
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
For any u ∈ U, let us denote (X ,Y ,Z ) the corresponding trajectory. Weintroduce the adjoint equations and the Hamiltonian function for thecontrol problem.
dpt = −Hx (t,Xt ,Yt , ut , pt ,Qt , kt) dt + ktdWt
pT = gx (XT ) QT(5.13)
anddQt = Hy (t,Xt ,Yt , ut , pt ,Qt , kt) dt +Hz (t,Xt ,Yt , ut , pt ,Qt , kt) dMX
t
Q0 = ly (Y0)(5.14)
The Hamiltonian function is defined by
H (t, x , y , q, p,Q, k) := h (t, x , y , u) + 〈p, b (t, x , u)〉+ 〈k, σ (t, x)〉+ 〈Q, f (t, x , y , u)〉 . (5.15)
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
Lemma 5.7. For any u, v ∈ U along with the corresponding trajectories(X ,Y ,Z ) , (X ′,Y ′,Z ′) and the solutions (p,Q, k) , (p′,Q ′, k ′) of thecorresponding adjoint equations, it hold that
E
(sup
t∈[0,T ]
|Qt − Q ′t |
2
)≤ C1 (d (u, v))
12 , (5.17)
and
E
∫ T
0
|pt − p′t |
2+ |kt − k ′t |
2
dt ≤ C1 (d (u, v))12 . (5.18)
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
A suitable version of Ekeland’s variational principle implies that, given asequence of positive real numbers εn > 0 with lim
n→+∞εn = 0, there exists
an admissible control un such that
J (un) ≤ infu∈U
J (u) + εn
andJ (un) ≤ J (v) + εnd (v , un) ;∀v ∈ U.
Remark 5.8. un which is εn-optimal for the cost J (u) is in fact optimalfor the new perturbed cost J (u) = J (u) + εnd (u, un) .
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
The next proposition gives necessary conditions for near optimalitysatisfied by the minimizing sequence (un).Proposition Let un be an εn − optimalcontrolthen, for any γ ∈
[0, 1
6
)there exists a constant C2 = C2 (γ) > 0 such that
E
∫ T
0
pns (b (s,X n
s , u)− b (s,X ns , u
ns )) + Qn
s (f (s,X ns ,Y
ns , u)− f (s,X n
s ,Yns , u
ns )) ds
+ E
∫ T
0
(h (s,X ns ,Y
ns , u)− h (s,X n
s ,Yns , u
ns )) ds ≥ −C2ε
γn ,∀u ∈ A.
(5.19)
where (pnt ,Q
nt , k
nt ) are the solutions of the adjoint equations
corresponding to (X n,Y n,Z n, un) .
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
Let q be a relaxed optimal control,(X , Y , Z
)be the corresponding
trajectory.Define the following adjoint processes dpt = −
∫AHx
(t, Xt , Yt , a, pt , Qt , kt
)qt (da) dt + ktdWt
pT = gx
(XT
)QT
(5.25)
anddQt =
∫AHy qt (da) dt +
∫AHz qt (da) dMX
t
Q0 = ly(Y0
),
(5.26)
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
Theorem ( The Pontryagin relaxed maximum principle). If(X , Y , Z , q
)denotes an optimal relaxed uplet, then there exists a
Lebesgue negligible subset N such that, for any t not in N,∫A
H(t, Xt , Yt , a, pt , Qt , kt
)qt (da) (5.29)
= supq∈P(A)
∫A
H(t, Xt , Yt , a, pt , Qt , kt
)qt (da) , P-a.s.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE
SUMMARYFormulation of the problem
The relaxed control problemExistence
The maximum principle
The maximum principle
The proof of the main result is based on the following stability lemma.
Lemma 5.15. Let pn,Qn, kn (resp. p, Q, k) be defined by (5.13), (5.14)(resp. (5.25), (5.26)), then , we have
i”) limn→+∞
E
(sup
0≤t≤T|pn
t − pt |2 +
∫ T
0
∣∣∣knt − kt
∣∣∣2 dt
)= 0,
ii”) limn→+∞
E
(sup
0≤t≤T
∣∣∣Qnt − Qt
∣∣∣2) = 0,
iii”) limn→+∞
∫ T
0
E (H (s,X nt ,Y
nt , u
nt , p
nt ,Q
nt , k
nt )) dt
=
∫ T
0
E(H(s, Xt , Yt , a, pt , Qt , kt
)qt (da)
)dt.
Brahim Mezerdi ON OPTIMAL CONTROL OF FBSDE