backward stochastic di erential equations and...
TRANSCRIPT
Backward Stochastic Differential Equations and
Applications in Finance
Ying Hu
August 10, 2013
1 Introduction
The aim of this short case is to present the basic theory of BSDEs and togive some applications in 2 different domains: mathematical finance andpartial differential equations.
This short course is divided into 3 parts: Part 1 gives the basic theoryof BSDEs; Part 2 gives the link between BSDEs and PDEs and Part 3 givesthe recent development of quadratic BSDEs.
To solve a BSDE, it is to find a couple of adapted processes (Y, Z)satisfying the following SDE:
dYt = −f(t, Yt, Zt)dt+ ZtdWt, 0 ≤ t ≤ T,
with the terminal condition (that’s why we call backward) YT = ξ, where ξis a square-integrable random variable. As SDEs, such equations should beunderstood in the sense of integral, i.e.
Yt = ξ +∫ T
tf(s, Ys, Zs)ds−
∫ T
tZsdWs, 0 ≤ t ≤ T.
The BSDEs were introduced in 1973 by J.M. Bismut in the case wheref is linear w.r.t. (Y, Z); and Pardoux and Peng developed the theory ofBSDEs where f is Lipschitz.
Since then, BSDEs have been extensively studied because of their pro-found applications to mathematical finance as well as the link with PDEs.Let us first give some examples.
1
In finance, an important question is to determine the price of an option.The simplest example is that of the Black-Scholes model and an Europeancall. The price of this financial produce Vt satisfies the equation:
dVt = (rVt + θZt)dt+ ZtdWt,
where r is the short term interest rate and θ is the risk premium, with theterminal condition VT = (ST −K)+ where St is the price of the stock andK is a constant, which is fixed in advance. We see that it is a linear BSDEin this simple model but it can be non-linear in more complicated financialmodels.
Let us look at a second example. Consider the following PDE:
∂tu(t, x) +12∂2x,xu(t, x) + f(u(t, x)) = 0, u(T, x) = g(x).
Let us suppose that this PDE admits a classical solution u. ApplyingIto’s formula to u(s,Ws); we obtain:
du(s,Ws) = ∂su(s,Ws) +12∂2x,xu(s,Ws)ds+ ∂xu(s,Ws)dWs
= −f(u(s,Ws))ds+ ∂xu(s,Ws)dWs.
We obtain still one BSDE - which is nonlinear if f is - Putting Ys =u(s,Ws) and Zs = ∂xu(s,Ws) because
dYs = −f(Ys)ds+ ZsdWs, with, YT = g(WT ).
2 Part 1: BSDE: Lipschitz Case
The aim of this section is to introduce the definition of BSDE, and to pre-cisely give the notations. Then we give the basic result in Lipschitz case:existence and uniqueness; linear BSDE and comparison theorem; and someexamples.
2.1 Vocabulary and notation
2.1.1 Presentation of the problem.
Let (Ω,F , (Ft)t≥0, P ) be a filtered probability space and ξ an FT -measurablerandom variable. One wants to solve the following ODE:
−dYtdt
= f(Yt), t ∈ [0, T ], with, YT = ξ,
2
moreover for any t, Yt does not depend on the future after t, i.e. the processY is (Ft)t≥0-adapted.
Take the simplest example f = 0. The natural candidate is Yt = ξwhich is not adapted if ξ is not deterministic. The best approximation sayin L2 - adapted is the martingale Yt = E(ξ|Ft). If we work with the naturalfiltration of a BM, the Brownian martingale representation theorem permitsto construct a square-integrable adapted process Z such that
Yt = E(ξ|Ft) = E[ξ] +∫ t
0ZsdWs.
One elementary calculus gives
Yt = ξ −∫ T
tZsdWs, i.e. dYt = ZtdWt, with, YT = ξ.
We see that the second unknown (the process Z) appears to let Y be adaptedin this simplest example.
As a generalization, we permit f to depend on the process Z; the equa-tion becomes then:
dYt = −f(t, Yt, Zt)dt+ ZtdWt, with, YT = ξ.
2.1.2 Notations.
Let (Ω,F , P ) a complete probability space and W a d-dimensional BM onthis space. We will denote by (Ft)t≥0 the natural filtration of the BM W .We will work with two spaces of processes:
• Denote by S2(Rk) the vector space formed by the processes Y , pro-gressively measurable, with values in Rk, such that
||Y ||2S2 = E[ sup0≤t≤T
|Yt|2] <∞,
and S2c the subspace formed by the continuous processes. Two indis-
tinguishable processes will be equal and we will use the same notationsfor the quotient spaces.
• Denote M2(Rk×d) as the set of those processes Z, progressively mea-surable, with values in Rk×d, such that
||Z||2M2 = E[∫ T
0||Zt||2dt] <∞,
3
where if z ∈ Rk×d, ||z||2 = trace(zz∗). Denote M2(Rk×d) as the set ofequivalent classes of M2(Rk×d).
Rk and Rk×d will be often omitted; the spaces S2, S2c and M2 are Banach
spaces for the defined norms. We denote by B2 the Banach space S2(Rk)×M2(Rk×d).
In this chapter, we are given one random mapping defined on [0, T ] ×Ω × Rk × Rk×d with values in Rk such that, for any (y, z) ∈ Rk × Rk×d,the process f(t, y, z)0≤t≤T is progressively measurable. We consider alsoa random variable ξ, FT -measurable and with values in Rk.
In this part, we want to solve the following BSDE:
dYt = −f(t, Yt, Zt)dt+ ZtdWt, 0 ≤ t ≤ T, YT = ξ,
Or, equivalently, in integral form:
Yt = ξ +∫ T
tf(r, Yr, Zr)dr −
∫ T
tZrdWr, 0 ≤ t ≤ T. (1)
The function f is called the generator of the BSDE and ξ the terminalcondition. Now, we can give a precise definition of the solution of BSDE(1).
Definition 1 One solution of the BSDE (1) is a couple of processes (Y, Z)satisfying:
1. Y and Z are progressively measurable with values respectively in Rk
and Rk×d;2. P − a.s.
∫ T0 (|f(r, Yr, Zr)|+ |Zr|2)dr <∞;
3. P − a.s., we have:
Yt = ξ +∫ T
tf(r, Yr, Zr)dr −
∫ T
tZrdWr, 0 ≤ t ≤ T.
Remark 2 It is important de note the following two points: first, the inte-grals of the equation (1) are well defined, Y is a continuous semi-martingale;then, as the process Y is progressively measurable, it is adapted and then inparticular Y0 is deterministic.
Before to give the first theorem of existence and uniqueness, we aregoing to show that, under some reasonable hypothesis on the generator f ,the process Y belongs to S2.
4
Proposition 3 Suppose that there exist a process f ∈ M2(R) and a con-stant λ such that:
∀(t, y, z) ∈ [0, T ]×Rk ×Rk×d, |f(t, y, z)| ≤ ft + λ(|y|+ |z|).
If (Y, Z) is a solution of the BSDE (1) such that Z ∈ M2, then Y belongsto S2
c .
Proof: The result follows from Gronwall’s inequality and the fact that Y0
is deterministic. In fact, we have, for any t ∈ [0, T ],
Yt = Y0 −∫ t
0f(r, Yr, Zr)dr +
∫ t
0ZrdWr,
and then, using the hypothesis on f ,
|Yt| ≤ |Y0|+∫ T
0(fr + λ|Zr|)dr + sup
0≤t≤T|∫ t
0ZrdWr|+ λ
∫ t
0|Yr|dr.
Setting
ζ = |Y0|+∫ T
0(fr + λ|Zr|)dr + sup
0≤t≤T|∫ t
0ZrdWr|.
From the hypothesis, Z belongs to M2 and then, via Doob’s inequality, thethird term is square-integrable ; it is the same for f , and Y0 is deterministichence square-integrable; we deduce that ζ is a square-integrable randomvariable.
As Y is a continuous process - see the last remark, Gronwall’s inequalitygives the inequality sup0≤t≤T |Yt| ≤ ζeλT which shows that Y belongs to S2.
Remark 4 The result remains true when ||f·||1 is a square-integrable ran-dom variable.
We finish by a result of integrability which will be used several times.
Lemma 5 Let Y ∈ S2(Rk) and Z ∈M2(Rk×d). Then∫ t
0 Ys · ZsdWs, t ∈ [0, T ]
is a uniformly integrable martingale.
Proof: The Burkholder-Davis-Gundy’s inequalities give
E
[sup
0≤t≤T
∣∣∣∣∫ t
0Ys · ZsdWs
∣∣∣∣]≤ CE
[(∫ T
0|Yr|2|Zr|2dr
)1/2]
≤ CE
[sup
0≤t≤T|Yt|
(∫ T
0|Zr|2dr
)1/2],
5
and then, as ab ≤ a2/2 + b2/2,
E
[sup
0≤t≤T
∣∣∣∣∫ t
0Ys · ZsdWs
∣∣∣∣]≤ C ′
(E
[sup
0≤t≤T|Yt|2
]+ E
[∫ T
0|Zr|2dr
]).
But the last quantity is finite by the hypothesis; we get the result.
2.2 Lipschitz case
2.2.1 Pardoux-Peng’s result
In this subsection, we will show the first result of existence and uniqueness.This result is due to E. Pardoux and S. Peng; this is the first result ofexistence and uniqueness for the BSDEs when the generator is non-linear.
Recall for the last time that f is defined on [0, T ]×Ω×Rk×Rk×d with val-ues in Rk such that, for any (y, z) ∈ Rk×Rk×d, the process f(t, y, z)0≤t≤Tis progressively measurable. We consider also ξ a random variable, FT -measurable, with valued in Rk.
Here are the hypotheses under which we will study:(L) There exists a constant λ such that P -a.s.,1. Condition of Lipschitz w.r.t. (y, z): for any t, y, y′, z, z′,
|f(t, y, z)− f(t, y′, z′)| ≤ λ(|y − y′|+ |z − z′|);
2. Condition of integrability:
E
[|ξ|2 +
∫ T
0|f(r, 0, 0)|2dr
]<∞.
We begin with a very simple case when f does depend neither on y noron z, i.e. we will be given square-integrable ξ and a process F in M2(Rk)and we want to find a solution of the BSDE
Yt = ξ +∫ T
tFrdr −
∫ T
tZrdWr, 0 ≤ t ≤ T. (2)
Lemma 6 Let ξ ∈ L2(FT ) and Ft0≤t≤T ∈ M2(Rk). The BSDE (2) ad-mits a unique solution (Y,Z) such that Z ∈M2.
Proof: Let us first suppose that (Y,Z) is a solution satisfying Z ∈M2. Ifwe take the conditional expectation w.r.t. Ft, we deduce
Yt = E
(ξ +
∫ T
tFrdr
∣∣∣Ft) .6
We define then Y with the help of the last formula, and it remainsto find Z. Note that because of Fubini’s theorem, as F is progressivelymeasurable,
∫ t0 Frdr is an (Ft)t∈[0,T ] adapted process; in fact in S2
c as F issquare-integrable integrable. We have then, for any t ∈ [0, T ],
Yt = E
(ξ +
∫ T
0Frdr
∣∣∣Ft)− ∫ t
0Frdr = Mt −
∫ t
0Frdr.
M is a Brownian martingale. By the Brownian martingale representationtheorem, we construct a process Z belonging to M2 such that
Yt = Mt −∫ t
0Frdr = M0 +
∫ t
0ZrdWr −
∫ t
0Frdr.
We check easily that (Y, Z) constructed is a solution of the BSDE becauseYT = ξ,
Yt − ξ = M0 +∫ t
0ZrdWr −
∫ t
0Frdr −
(M0 +
∫ T
0ZrdWr −
∫ T
0Frdr
)=
∫ T
tFrdr −
∫ T
tZrdWr.
The uniqueness is evident for the solutions satisfying Z ∈M2.We show now the Pardoux-Peng Theorem.
Theorem 7 [Pardoux-Peng 90] Under the hypothesis (L), the BSDE (1)admits a unique solution (Y, Z) such that Z ∈M2.
Proof: We use a fixed point argument in the Banach spaceB2 by construct-ing an application Ψ from B2 in itself such that (Y,Z) ∈ B2 is solution ofthe BSDE (1) if and only if it is a fixed point of Ψ.
For (U, V ) element of B2, we define (Y,Z) = Ψ(U, V ) as being the solu-tion of the BSDE:
Yt = ξ +∫ T
tf(r, Ur, Vr)dr −
∫ T
tZrdWr, 0 ≤ t ≤ T.
Note that this last BSDE admits a unique solution which is in B2. Infact, set Fr = f(r, Ur, Vr). This process belongs to M2 because, f beingLipschitz,
|Fr| ≤ |f(r, 0, 0)|+ λ|Ur|+ λ|Vr|,
7
and the last three processes are square-integrable. The, we can apply theLemma 6 to obtain a unique solution (Y,Z) ∈ B2. The mapping Ψ from B2
in itself is then well defined.Let (U, V ) and (U ′, V ′) two elements in B2 and (Y, Z) = Ψ(U, V ),
(Y ′, Z ′) = Ψ(U ′, V ′). Denote y = Y − Y ′ and z = Z − Z ′. We have,yT = 0 and
dyt = −f(t, Ut, Vt)− f(t, U ′t , V′t )dt+ ztdWt.
We apply Ito’s formula to eαt|yt|2 to obtain
d(eαt|yt|2) = αeαt|yt|2dt−2eαtyt·f(t, Ut, Vt)−f(t, U ′t , V′t )dt+2eαtyt·ztdWt+eαt|zt|2dt.
As a consequence, integrating from t to T , we obtain
eαt|yt|2 +∫ T
teαr|zr|2dr =
∫ T
teαr(−α|yr|2 + 2yr · f(r, Ur, Vr)− f(r, U ′r, V
′r )dr
−∫ T
t2eαryr · zrdWr,
and, as f is Lipschitz, it follows, denoting by u and v for U −U ′ and V −V ′respectively,
eαt|yt|2+∫ T
teαr|zr|2dr ≤
∫ T
teαr(−α|yr|2+2λ|yr| |ur|+2λ|yr| |vr|)dr−
∫ T
t2eαryr·zrdWr.
For any ε > 0, we have 2ab ≤ a2/ε + εb2, and then, the last inequalitygives
eαt|yt|2 +∫ T
teαr|zr|2dr ≤
∫ T
teαr(−α+ 2λ2/ε)|yr|2dr −
∫ T
t2eαryr · zrdWr
+ε∫ T
teαr(|ur|2 + |vr|2)dr,
and putting α = 2λ2/ε, we have, denoting Rε = ε∫ T0 eαr(|ur|2 + |vr|2)dr,
∀t ∈ [0, T ], eαt|yt|2 +∫ T
teαr|zr|2dr ≤ Rε − 2
∫ T
teαryr · zrdWr. (3)
From the Lemma 5, the local martingale ∫ t0 e
αryr · zrdWrt∈[0,T ] is infact a real martingale null at 0 as Y, Y ′ belong to S2 and Z,Z ′ belong M2.
8
In particular, taking the expectation – the stochastic integral disappearsvia the last remark – we obtain easily, for t = 0, that
E
[∫ T
0eαr|zr|2dr
]≤ E[Rε]. (4)
Coming back to the inequality (3), the BDG’s inequalities provide – withC universal –
E
[sup
0≤t≤Teαt|yt|2
]≤ E[Rε] + CE
[(∫ T
0e2αr|yr|2|zr|2dr
)1/2]
≤ E[Rε] + CE
[sup
0≤t≤Teαt/2|yt|
(∫ T
0eαr|zr|2dr
)1/2],
then, as ab ≤ a2/2 + b2/2,
E
[sup
0≤t≤Teαt|yt|2
]≤ E[Rε] +
12E
[sup
0≤t≤Teαt|yt|2
]+C2
2E
[∫ T
0eαr|zr|2dr
].
Taking into consideration the inequality (4), we obtain finally
E
[sup
0≤t≤Teαt|yt|2 +
∫ T
0eαr|zr|2dr
]≤ (3 + C2)E[Rε],
and then, coming back to the definition of Rε,
E
[sup
0≤t≤Teαt|yt|2 +
∫ T
0eαr|zr|2dr
]≤ ε(3+C2)(1∨T )E
[sup
0≤t≤Teαt|ut|2 +
∫ T
0eαr|vr|2dr
].
Taking ε such that ε(3 + C2)(1 ∨ T ) = 1/2, the mapping Ψ is a strictcontraction from B2 to itself if we equipped it with the norm
||(U, V )||α = E
[sup
0≤t≤Teαt|Ut|2 +
∫ T
0eαr|Vr|2dr
]1/2
,
which is a Banach space - this last norm being equivalent to the usual normcorresponding to the case when α = 0.
Ψ has then a unique fixed point , which ensure the existence and theuniqueness of a solution of the BSDE (1) in B2.
We obtain then a unique solution satisfying Z ∈M2 because the Propo-sition 3 implies that such a solution belongs to B2.
Remark 8 From now on, the expression “the solution of the BSDE” signi-fies the solution of the BSDE satisfying Z ∈M2.
9
2.2.2 The role of Z
We are going to see that the role of Z, more precisely that of term∫ Tt ZrdWr
is to render the process Y adapted and when this is not necessary, Z is zero.
Proposition 9 Let (Y,Z) the solution of the (1) and let τ a stopping timeup bounded by T . We suppose, besides the hypothesis (L), that ξ is Fτ -measurable and that f(t, y, z) = 0 as soon as t ≥ τ .
Then Yt = Yt∧τ and Zt = 0 if t ≥ τ .
Proof: We have, P-a.s.,
Yt = ξ +∫ T
tf(r, Yr, Zr)dr −
∫ T
tZrdWr, 0 ≤ t ≤ T,
and then, for t = τ , as f(t, y, z) = 0 as soon as t ≥ τ ,
Yτ = ξ +∫ T
τf(r, Yr, Zr)dr −
∫ T
τZrdWr = ξ −
∫ T
τZrdWr.
As a consequence Yτ = E(ξ|Fτ ) = ξ and then∫ Tτ ZrdWr = 0 from which
we deduce that
E
[(∫ T
τZrdWr
)2]
= E
[∫ T
τ|Zr|2dr
]= 0,
and finally that Zr1r≥τ = 0.It follows immediately that, if t ≥ τ , Yt = Yτ , as from the hypothesis,
Yτ = Yt +∫ t
τf(r, Yr, Zr)dr −
∫ t
τZrdWr = Yt,
which concludes the proof.Note that when ξ and f are deterministic, Z is zero and Y is the solution
of the ODEdYtdt
= −f(t, Yt, 0), YT = ξ.
2.2.3 A priori estimate
We finish this section by giving a first estimate on the BSDEs: it is in factto study the dependance of the solution of the BSDE w.r.t. the parametersξ and the process f(t, 0, 0)0≤t≤T .
10
Proposition 10 Suppose that (ξ, f) satisfies (L). Let (Y, Z) the solution ofthe BSDE (1) such that Z ∈M2. Then, there exists a constant Cu universalsuch that, for any β ≥ 1 + 2λ+ 2λ2,
E
[sup
0≤t≤Teβt|Yt|2 +
∫ T
0eβt|Zt|2dt
]≤ CuE
[eβT |ξ|2 +
∫ T
0eβt|f(t, 0, 0)|2dt
].
Proof: We apply Ito’s formula to eβt|Yt|2 to obtain:
eβt|Yt|2+∫ T
teβr|Zr|2dr = eβT |ξ|2+
∫ T
teβr(−β|Yr|2+2Yr·f(r, Yr, Zr))dr−
∫ T
t2eβrYr·ZrdWr.
As f is λ-Lipschitz, we have, for any (t, y, z),
2y · f(t, y, z) ≤ 2|y| |f(t, 0, 0)|+ 2λ|y|2 + 2λ|y| |z|,
and then using the fact that 2ab ≤ εa2 + b2
ε for ε = 1 and ε = 2, we have
2y · f(t, y, z) ≤ (1 + 2λ+ 2λ2)|y|2 + |f(t, 0, 0)|2 + |z|2/2.
For β ≥ 1 + 2λ+ 2λ2, we obtain, for any t ∈ [0, T ],
eβt|Yt|2+12
∫ T
teβr|Zr|2dr ≤ eβT |ξ|2+
∫ T
0eβr|f(r, 0, 0)|2dr−2
∫ T
teβrYr·ZrdWr.
(5)The local martingale
∫ t0 e
βrYr · ZrdWr, t ∈ [0, T ] is a martingale - cf.Lemma 5. In particular, taking the expectation, we obtain easily, for t = 0,
E
[∫ T
0eβr|Zr|2dt
]≤ 2E
[eβT |ξ|2 +
∫ T
0eβr|f(r, 0, 0)|2dr
].
Coming back to the inequality (5), the BDG’s inequalities provide - with Cuniversal -,
E
[sup
0≤t≤Teβt|Yt|2
]≤ E
[eβT |ξ|2 +
∫ T
0eβr|f(r, 0, 0)|2dr
]+CE
[(∫ T
0e2βr|Yr|2|Zr|2dr
)1/2].
On the other hand,
CE
[(∫ T
0e2βr|Yr|2|Zr|2dr
)1/2]≤ CE
[sup
0≤t≤Teβt/2|Yt|
(∫ T
0eβr|Zr|2dr
)1/2]
≤ 12E
[sup
0≤t≤Teβt|Yt|2
]+C2
2
[∫ T
0eβr|Zr|2dr
].
11
It follows that,
E
[sup
0≤t≤Teβt|Yt|2
]≤ 2E
[eβT |ξ|2 +
∫ T
0eβr|f(r, 0, 0)|2dr
]+C2E
[∫ T
0eβr|Zr|2dr
],
and finally we obtain
E
[sup
0≤t≤Teβt|Yt|2 +
∫ T
0eβt|Zt|2dt
]≤ 2(2+C2)E
[eβT |ξ|2 +
∫ T
0eβt|f(t, 0, 0)|2dt
],
which concludes the proof of the proposition by taking Cu = 2(2 + C2).
2.3 Linear BSDE and comparison theorem
2.3.1 linear BSDE
In this subsection, we study particular cases of linear BSDEs for which wegive some explicit formulae.
Let k = 1; Y is real and Z is a row vector of dimension d.
Proposition 11 Let (at, bt) be a process with values in R×Rd, progressivelymeasurable and bounded. Let ct be an element in M2(R) and ξ a randomvariable, FT -measurable, square-integrable with real values. Then the linearBSDE
Yt = ξ +∫ T
tarYr + Zrbr + crdr −
∫ T
tZrdWr,
admits a unique solution:
Yt = Γ−1t E
(ξΓT +
∫ T
tcrΓrdr|Ft
),
with, for any t ∈ [0, T ],
Γt = exp∫ t
0br · dWr −
12
∫ t
0|br|2dr +
∫ t
0ardr
.
Proof: We begin by noting that the process Γ satisfies:
dΓt = Γt(atdt+ bt · dWt), Γ0 = 1.
On the other hand, as b is bounded, Doob’s inequality shows that Γ belongsto S2.
12
Furthermore, the hypotheses of this proposition ensures the existence ofa unique solution (Y, Z) to the linear BSDE; it suffices to set f(t, y, z) =aty + zbt + ct and to check that (L) is satisfied. Y belongs to S2 from theProposition (3).
The integration by parts formula gives
dΓtYt = ΓtdYt + YtdΓt + d〈Γ, Y 〉t = −Γtctdt+ ΓtZtdWt + ΓtYtbt · dWt,
which shows that the process ΓtYt +∫ t0 csΓsds is a local martingale which is
in fact a martingale as c ∈M2 and Γ, Y are in S2. It follows that
ΓtYt +∫ t
0crΓrdr = E
(ΓTYT +
∫ T
0crΓrdr|Ft
),
which gives the announced formula.
Remark 12 Note that if ξ ≥ 0 and ct ≥ 0 then the solution of the linearBSDE verifies Yt ≥ 0.
This remark permits us to obtain the comparison theorem in the nextsubsection. To illustrate this result taking the case when a and c are zero.We have then
Yt = E
(exp
∫ T
tbr · dWr −
12
∫ T
t|br|2dr
ξ|Ft
)= E∗(ξ|Ft),
where P ∗ is the density measure w.r.t. P
LT = exp∫ T
0br · dWr −
12
∫ T
0|br|2dr
.
Another method to see this, by the idea of risk-neutral probability, is tostudy the BSDE under P ∗. In fact, under P ∗, Bt = Wt −
∫ t0 brdr is a BM
by Girsanov theorem. Then the equation can be written
dYt = −Ztbtdt+ ZtdWt = ZtdBt, YT = ξ.
Then, under P ∗, Y is a martingale, which shows also the formula.
2.3.2 Comparison theorem
This subsection is devoted to the comparison theorem which permits tocompare the solutions of two BSDEs (in R) as soon as one can compare theterminal conditions and the generators.
13
Theorem 13 Suppose that k = 1 and that (ξ, f), (ξ′, f ′) verify the hy-pothesis (L). We denote (Y,Z) and (Y ′, Z ′) the solutions of correspondingBSDEs. We suppose also that P − a.s. ξ ≤ ξ′ and that f(t, Yt, Zt) ≤f ′(t, Yt, Zt) m⊗ P − a.e. (m is the Lebesgue measure). Then
P − a.s., ∀t ∈ [0, T ], Yt ≤ Y ′t .
Moreover, if Y0 = Y ′0, then P − a.s., Yt = Y ′t , 0 ≤ t ≤ T and f(t, Yt, Zt) =f ′(t, Y ′t , Z
′t) m ⊗ P − a.e. In particular, as soon as P (ξ < ξ′) > 0 or
f(t, Yt, Zt) < f ′(t, Yt, Zt) on a set with positive m⊗P -measure then Y0 < Y ′0.
Proof: We use the technique of liberalization which permits us to the linearBSDE case. We look for an equation satisfied by U = Y ′ − Y . DenotingV = Z ′ − Z and ζ = ξ′ − ξ,
Ut = ζ +∫ T
t(f ′(r, Y ′r , Z
′r)− f(r, Yr, Zr))dr −
∫ T
tVrdWr.
We divide the increments of f into three parts by writing
f ′(r, Y ′r , Z′r)− f(r, Yr, Zr) = f ′(r, Y ′r , Z
′r)− f ′(r, Yr, Z ′r) + f ′(r, Yr, Z ′r)− f ′(r, Yr, Zr)
+f ′(r, Yr, Zr)− f(r, Yr, Zr)(which is non-negative).
We introduce two processes a and b: a is with real values and b is acolumn vector of dimension d. We put
ar =f ′(r, Y ′r , Z
′r)− f ′(r, Yr, Z ′r)Ur
1|Ur|>0,
and
br =(f ′(r, Yr, Z ′r)− f ′(r, Yr, Zr))V ∗r
|Vr|21|Vr|>0.
We note that, as f ′ is Lipschitz, these two processes are progressively mea-surable and bounded. With these notations, we have:
Ut = ζ +∫ T
t(arUr + Vrbr + cr)dr −
∫ T
tVrdWr,
where cr = f ′(r, Yr, Zr) − f(r, Yr, Zr). From the hypothesis, we have ζ ≥ 0and cr ≥ 0. Applying the explicit formula for the linear BSDE - Proposition11)-, we have, for t ∈ [0, T ],
Ut = Γ−1t E
(ζΓT +
∫ T
tcrΓrdr|Ft
)14
where, for 0 ≤ r ≤ T ,
Γr = exp∫ r
0bu · dWu −
12
∫ r
0|bu|2du+
∫ r
0audu
.
As already mentioned in the remark following the Proposition 11, thisformula shows that Ut ≥ 0, as soon as ζ ≥ 0 and cr ≥ 0.
For the second part of the result (strict comparison), if moreover, U0 = 0we have
0 = E
(ζΓT +
∫ T
0crΓrdr|Ft
),
which concludes the proof.
2.3.3 Black-Scholes model
The Black-Scholes model is an example of linear BSDE.In a financial market there exists one stock whose price is given by the
SDEdSt = St(µdt+ σdWt), S0 = x,
where µ ∈ R, σ > 0; the parameter σ is called the volatility. We have, forany t ≥ 0,
St = x expσWt + (µ− σ2/2)t.
There exists also a risk-free asset, whose price is given by
dEt = rEtdt, E0 = y, i.e. Et = yert.
The strategy is given by a couple of processes, (pt, qt), which are adaptedprocesses. pt is the number of stock, and qt the number of risk-free asset. Ifwe consider only the self-financing strategy, then the value of the portfoliowill be given by
dVt = qtdEt + ptdSt = rqtEtdt+ ptSt(µdt+ σdWt).
As qtEt = Vt − ptSt, and then, we have by noting πt = ptSt(the amount ofmoney invested in risky asset),
dVt = rVtdt+ πtσ(µ− r)/σdt+ πtσdWt.
Putting Zt = πtσ and θ = (µ− r)/σ (the risk premium),
dVt = rVtdt+ θZtdt+ ZtdWt.
15
One problem in finance is to give a price to options. A European buy-option (call option) with maturity T and exercise price K is a contractwhich gives the right but no obligation to his holder to buy one share of thestock at the exercise price K. Equivalently, the seller of the option has topay (ST − K)+ to its holder, which represents the profit that permits theexercise of the option. More generally we can imagine a claim whose profitis just a non-negative random variable ξ depending on S. At which price vshould one sell the option?
To find v, the fundamental idea is the replication, the seller sells theoption at the price v and invests this sum in the market following the strategyZ to be found. The value of his portfolio is governed by the SDE:
dVt = rVtdt+ θZtdt+ ZtdWt, V0 = v.
The problem is then to find v and Z such that the solution of the previousSDE verifies VT = ξ; we say that in this case v is the fair price. i.e., can wefind adapted (V,Z) such that
dVt = rVtdt+ θZtdt+ ZtdWt, VT = ξ.
In this case it suffices to sell the option at the price v = V0. Hence thepricing is to solve the BSDE which is linear.
Suppose now that the regulator of the market wants to avoid short-sellingof the stock. It can dissuade this kind of transaction by penalizing investorsby a proportional cost βπ−t = γZ−t (γ > 0). In this case, replicating a claimit to solve the following BSDE
dVt = (rVt + θZt − γZ−t )dt+ ZtdWt, VT = ξ.
This BSDE is not linear but it verifies still the hypothesis (L). Anotherexample of BSDE nonlinear encountered in finance is the following
dVt = (rVt + θZt)dt+ ZtdWt − (R− r)(Vt − Zt/σ)−dt, VT = ξ.
We should solve this last BSDE to replicate a claim when the borrowingrate R is bigger than the lending rate r.
We remark that the strategies are admissible (Vt ≥ 0). This result followsfrom the comparison theorem as ξ ≥ 0 and f(t, 0, 0) ≥ 0.
16
3 Part 2: BSDE and PDE
We consider here the Markovian BSDEs: it is a very special case when thethe terminal condition and the generator on ω is given via the solution of aSDE. We will see that the Markov property holds for the BSDE as soon asit holds for the SDE, that is why we call it as Markovian case.
3.1 Elementary properties of flow
We will work here for SDEs with deterministic initial conditions which per-mits us to take as filtration the natural filtration of BM FWt t≥0. We con-sider two continuous functions b : [0, T ] × Rn → Rn and σ : [0, T ] × Rn →Rn×d. We suppose that there exists one constant K > 0 such that, for anyt, for any x, x′ in Rn,
1. |b(t, x)− b(t, x′)|+ |σ(t, x)− σ(t, x′)| ≤ K|x− x′|;
2. |b(t, x)|+ |σ(t, x)| ≤ K(1 + |x|).Under these hypotheses, we can construct, given t ∈ [0, T ] and a random
variable θ ∈ L2(Ft), Xt,θu t≤u≤T as the solution of the SDE:
Xt,θu = θ +
∫ u
tb(r,Xt,θ
r )dr +∫ u
tσ(r,Xt,θ
r )dWr, t ≤ u ≤ T ; (6)
as a convention, if 0 ≤ u ≤ t, Xt,θu = E(θ|Fu).
In the deterministic case, if σ = 0, the flow Xt,xs , denoted φt,xs , have
many properties. In particular:1.φt,xs is Lipschitz w.r.t. (t, x, s);2. for r ≤ t ≤ s, φr,xs = φ
t,φr,xt
s ;3. if t ≤ s, x→ φt,xs is a homeomorphism of Rn.
3.1.1 Continuity
To show the continuity properties of the flow, we will apply the criterion ofKolmogorov. In order to do it, we should establish some estimates on themoments of Xt,x
s . The proofs are quite involved but not very difficult: itsuffices to apply Holder’s inequality and Gronwall’s lemma.
Proposition 14 Let p ≥ 1. There exists a constant C, depending on T andp only, such that:
∀t ∈ [0, T ], ∀x ∈ Rn, E
[sup
0≤s≤T|Xt,x
s |p]≤ C(1 + |x|p) (7)
17
We know that the solution of the SDE has the moments of all orders. Byclassical results, we also have an estimate of the same type for the momentsof increments of X.
Proposition 15 Let 2 ≤ p < ∞. There exists a constant C such that, forany (t, x), (t′, x′) belonging to [0, T ]×Rn,
E
[sup
0≤s≤T|Xt,x
s −Xt′,x′s |p
]≤ C(|x− x′|p + |t− t′|p/2(1 + |x′|p)). (8)
Corollary 16 There exists a modification of the process X such that themapping from [0, T ]×Rn with values in S2
c , (t, x)→ (s→ Xt,xs ) is continu-
ous. In particular, (t, x, s)→ Xt,xs is continuous.
Proof: This is a direct application of the previous estimate which is truefor any p ≥ 2 and of the criterion of Kolmogorov.
Remark 17 It is important to note that the previous corollary implies inparticular that, P-a.s., for any (t, x, s), the equation (6) is true.
We end this subsection by precisely indicating the regularity of the flowgenerated by the SDE.
Proposition 18 Let 2 ≤ p < ∞. There exists a constant C such that, forany (t, x, s), (t′, x′, s′),
E[|Xt,x
s −Xt′,x′
s′ |p]≤ C(|x− x′|p + (1 + |x′|p)(|t− t′|p/2 + |s− s′|p/2)). (9)
In particular, the trajectories (t, x, s)→ Xt,xs are Holder-continuous (locally
in x) in t of order β, in x of order α, and in s of order β, for any β < 1/2and α < 1.
Proof: The regularity of the trajectories follows from the criterion of Kol-mogorov and the estimate.
3.1.2 Markov Property
We will establish the Markov property for the solutions of the SDE (6) as aconsequence of the property of the flow.
Proposition 19 For any x ∈ Rn and any 0 ≤ r ≤ t ≤ s, we have
Xr,xs = X
t,Xr,xt
s , P − a.s.
18
Remark 20 In fact, by continuity, the inequality Xr,xs = X
t,Xr,xt
s happensfor any x and for any 0 ≤ r ≤ t ≤ s ≤ T except P -null set.
We will now establish the Markov property. Recall that a process X isMarkov if it does not depend on the past only given the present. We havethe following mathematical definition:
Definition 21 [Markov property] Let X a progressively measurable processw.r.t. the filtration Ftt≥0. We say that X has the Markov property w.r.t.Ftt≥0 if, for any bounded Borel function f , and for any t ≤ s,
E(f(Xs)|Ft) = E(f(Xs)|Xs), P − a.s.
Let us show the Markov property for the solutions of the SDE (6) w.r.t.the σ-algebra of the BM W .
Theorem 22 Let x ∈ Rn and t ∈ [0, T ]. (Xt,xs )0≤s≤T is a Markov process
w.r.t. the filtration of the BM. If f is measurable and bounded, then for anyt ≤ r ≤ s,
E(f(Xt,xs )|Fr) = Λ(Xt,x
r ) P − a.s.
with Λ(y) = E[f(Xr,ys )].
Proof: Let 0 ≤ t ≤ r ≤ s ≤ T . From the property of the flow, we have
Xt,xs = Xr,Xt,x
rs .
Moreover, we will show that Xr,ys is measurable w.r.t. the σ-algebra of the
increments of the BM σWr+u −Wr, u ∈ [0, s− r]. Hence,
Xr,ys = Φ(y,Wr+u −Wr; 0 ≤ u ≤ s− r),
where Φ is measurable. It follows that,
Xt,xs = Xr,Xt,x
rs = Φ(Xt,x
r ,Wr+u −Wr; 0 ≤ u ≤ s− r).
To conclude, we should note that Xt,xr is Fr-measurable and that Fr is
independent of the σ-algebra σWr+u −Wr, u ∈ [0, s− r]. We have then
E(f(Xt,xs )|Fr) = E(fΦ(Xt,x
r ,Wr+u −Wr; 0 ≤ u ≤ s− r)|Fr)= E [fΦ(y,Wr+u −Wr; 0 ≤ u ≤ s− r)]|y=Xt,x
r
= E [f(Xr,ys )]|y=Xt,x
r
= Λ(Xt,xr ).
19
which shows the desired property.We can also show that, if g is measurable and – for example – bounded,
then
E
(∫ T
rg(u,Xt,x
u )du|Fr)
= Γ(r,Xt,xr ),
where Γ(r, y) = E[∫ Tr g(u,Xr,y
u )du].
Remark 23 If b and σ does not depend on time – we say that the SDE ishomogenous – we can show that the law of Xt,x
s is the same as that X0,xt−s.
We have then
E(f(Xt,xs )|Fr) = E[f(X0,y
t−r)]|y=Xt,xr
P − a.s.
3.2 The model
3.2.1 Hypotheses and notations
Let us give a complete probability space (Ω,F , P ) on which is defined a d-dim BM W . The filtration Ftt≥0 is the natural filtration of W augmentedsuch that the usual hypotheses are satisfied.
We consider two continuous functions b : [0, T ] × Rn → Rn and σ :[0, T ] × Rn → Rn×d. We suppose that there exists a constant K > 0 suchthat, for any t and any x, x′ in Rn,
1.|b(t, x)− b(t, x′)|+ |σ(t, x)− σ(t, x′)| ≤ K|x− x′|;2.|b(t, x)|+ |σ(t, x)| ≤ K(1 + |x|).Under these hypotheses, we can construct, given a real t ∈ [0, T ] and a
random variable θ ∈ L2(Ft), Xt,θu t≤u≤T as the solution of the SDE
Xt,θu = θ +
∫ u
tb(r,Xt,θ
r )dr +∫ u
tσ(r,Xt,θ
r )dWr, t ≤ u ≤ T. (10)
As a convention, we define Xt,θu = E(θ|Fu) for 0 ≤ u ≤ t. The properties
Xt,θu t≤u≤T have been studied in the last subsection, in particular in the
case θ = x ∈ Rn.Consider also two continuous functions g : Rk → Rk and f : [0, T ]×Rn×
Rk×Rk×d → Rk. We suppose that f and g verify the following hypotheses:there exist two reals µ and p ≥ 1 such that, for any (t, x, y, x′, y′, z, z′),
1. |f(t, x, y, z)− f(t, x′, y′, z′)| ≤ µ|y − y′|;
2. |f(t, x, y, z)− f(t, x, y, z′)| ≤ K|z − z′|;
20
3. |g(x)|+ |f(t, x, y, z)| ≤ K(1 + |x|p + |y|+ |z|).
Under these hypotheses, if θ belongs to L2p(Ft), we can solve the BSDE
Y t,θu = g(Xt,θ
T ) +∫ T
uf(r,Xt,θ
r , Y t,θr , Zt,θr )dr −
∫ T
uZt,θr dWr, 0 ≤ u ≤ T.
(11)In this part, we will suppose that the previous hypotheses on the coef-
ficients b,σ, f and g are satisfied. Sometimes, we will suppose a strongerhypothesis on f and g. For example, we may use the (Lip) hypothesis, whereg is K-Lipschitz and f is K-Lipschitz in x uniformly w.r.t. (t, y, z).
We give two elementary properties of the BSDE (11).
Proposition 24 If θ ∈ L2p(Ft), then the BSDE (11) has a solution (veri-fying Z ∈M2), (Y t,θ
u , Zt,θu )0≤u≤T . Furthermore, there exists a constant Csuch that, for any t, for any θ ∈ L2p(Ft),
E
[sup
0≤u≤T|Y t,θu |2 +
∫ T
0|Zt,θr |2dr
]≤ C(1 + E[|θ|2p]).
Remark 25 In particular, when θ is a constant equal to x, we have
E
[sup
0≤u≤T|Y t,xu |2 +
∫ T
0|Zt,xr |2dr
]≤ C(1 + |x|2p).
Proposition 26 If θn converges to θ in L2p(Ft), then
E
[sup
0≤u≤T|Y t,θu − Y t,θn
u |2 +∫ T
0|Zt,θr − Zt,θn
r |2dr
]→ 0, if n→∞.
Moreover, under the hypothesis (Lip), we have, if θ and θ′ are Ft-measurableand square-integrable,
E
[sup
0≤u≤T|Y t,θu − Y t,θ′
u |2 +∫ T
0|Zt,θr − Zt,θ
′r |2dr
]≤ CE
[|θ − θ′|2
].
3.3 Markov Property of BSDE
In this subsection, we will establish that the Markov property of the solutionsof SDE translate to solutions of BSDE. This property is important for theapplications to PDE. We begin by show that Y t,x
t is a deterministic quantity.Introduce a new notation: if t ≤ u, F tu = σ(N ,Wr −Wt, t ≤ r ≤ u).
21
Proposition 27 If (t, x) ∈ [0, T ]×Rn, then (Xt,xu , Y t,x
u )t≤u≤T is adaptedw.r.t. the filtration F tut≤u≤T . In particular, Y t,x
t is deterministic.We can choose a version of Zt,xu t≤u≤T adapted w.r.t. the filtration
F tut≤u≤T .
Since Y t,xt is deterministic, we can define a function
∀(t, x) ∈ [0, T ]×Rn, u(t, x) := Y t,xt .
Let us begin by studying the growth and the continuity of this function.
Proposition 28 The function u is continuous and with polynomial growth,i.e.,
∀(t, x) ∈ [0, T ]×Rn, |u(t, x)| ≤ C(1 + |x|p).
Moreover, under the hypothesis (Lip), the function u verifies, for any(t, x), (t′, x′), that
|u(t, x)− u(t′, x′)| ≤ C(|x− x′|+ |t− t′|1/2(1 + |x|)).
With the help of this function, we can establish the Markov property forthe BSDEs.
Theorem 29 Let t ∈ [0, T ] and θ ∈ L2p(Ft). We have:
P − a.s. Y t,θt = u(t, θ) = Y t,·
t θ.
Corollary 30 Let t ∈ [0, T ] and θ ∈ L2p(Ft). Then, we have P -a.s.,
∀s ∈ [t, T ], Y t,θs = u(s,Xt,θ
s ).
Remark 31 We use often this result with θ constantly equal to x. ThisMarkov property, Y t,x
s = u(s,Xt,xs ), plays an important role when we try
to construct the solution of the PDE with the help of BSDE: we will see itlately.
3.4 Feynman-Kac’s Formula
We finish this part with the link between the BSDEs and the PDEs. In thelast subsection, we have seen that Y t,x
r = u(r,Xt,xr ) where u is a deterministic
function. We are going to see that u is solution of a PDE. We will note Lthe following second order differential operator : for h regular
Lh(t, x) =12
trace(σσ∗(t, x)D2h(t, x)) + b(t, x) · ∇h(t, x).
22
Concerning the notations, if h is a function of t and x we denote ∂th or h′
the partial differential in time, ∇h the gradient in space ( column vector)and Dh = (∇h)∗, D2h the matrix of second-order differential.
We suppose also that the process Y is real, i.e., k = 1. Z is a line vectorof dimension d (that of BM). To sum up, f : [0, T ]×Rn ×R×R1×d → R.
The aim of this part is to establish the relations between Y t,xr , Zt,xr 0≤r≤T
solution of the BSDE
Y t,xr = g(Xt,x
T ) +∫ T
rf(u,Xt,x
u , Y t,xu , Zt,xu )du−
∫ T
rZt,xu dWu, 0 ≤ r ≤ T,
(12)where Xt,x
r 0≤r≤T is a solution of the SDE - with the convention Xt,xr = x
if 0 ≤ r ≤ t -
Xt,xr = x+
∫ r
tb(u,Xt,x
u )du+∫ r
tσ(u,Xt,x
u )dWu, t ≤ r ≤ T, (13)
on the one hand and << the solution >> v of the following parabolic PDE:
∂th+Lv(t, x)+f(t, x, v(t, x), Dv(t, x)σ(t, x)) = 0, (t, x) ∈]0, T [×Rn, v(T, ·) = g.(14)
Proposition 32 Suppose that the PDE (14) has a solution v, of classC1,2([0, T ] × Rn), such that for any (t, x) ∈ [0, T ] × Rn, |∇v(t, x)| ≤C(1 + |x|q).
Then, for any (t, x) ∈ [0, T ] × Rn, the solution of the BSDE (13),Y t,x
r , Zt,xr t≤r≤T , is given by the couple of processes v(r,Xt,xr ), Dvσ(r,Xt,x
r )t≤r≤T .In particular, we obtain the formula,
u(t, x) = Y t,xt = v(t, x).
The previous result gives a probabilistic representation formula for thesolution of a nonlinear parabolic PDE - we say semi-linear because the non-linearity is not so strong -. This type of formulas, known under the nameof Feynman-Kac - following the works of Richard Feynman and Mark Kac -applies from the origin to the linear problems. We obtain as a corollary:
Corollary 33 Take f(t, x, y, z) = c(t, x)y + h(t, x), where c and h are twobounded continuous functions. Under the previous hypotheses , for any(t, x) ∈ [0, T ]×Rn,
v(t, x) = E
[g(Xt,x
T ) exp(∫ T
tc(r,Xt,x
r )dr)
+∫ T
th(r,Xt,x
r ) exp(∫ r
tc(s,Xt,x
s )ds)dr
].
23
Proof: According to the previous proposition, we know that v(t, x) = Y t,xt .
But when the function f takes the form f(t, x, y, z) = c(t, x)y+h(t, x), TheBSDE which we should solve is a linear BSDE. We use then the explicitformula to conclude.
Remark 34 In the last two results, we have supposed the existence of aclassical solution v and we have deduced the solution of BSDE and the for-mula v(t, x) = Y t,x
t . If the coefficients are regular, we can prove that u whichis defined by u(t, x) = Y t,x
t is a regular function which is a solution of thePDE. (14).
The previous method consists to study the PDE, then to deduce thesolutions of the BSDE. But we can also study the BSDE and deduce theconstruction of the solution of the PDE without assuming the regular coef-ficients. For this we apply the notion of the solution of viscosity of PDEs.
Let us recall the definition of viscosity solution.
Definition 35 A continuous function u, with u(T, ·) = g, is a viscosity sub-solution (supersolution) if, whenever u−φ has a local maximum (minimum)at (t, x) where φ is C1,2,
∂tφ(t, x) + Lφ(t, x) + f(t, x, u(t, x),∇φσ(t, x)) ≥ 0, (≤ 0)
A solution is both a sub and a supersolution.
Theorem 36 The function u(t, x) := Y t,xt is a solution of viscosity of the
PDE (14).
Proof: By construction u is continuous and u(T, ·) = g.Let us show that u is a subsolution. Let (t, x) ∈ [0, T ) × Rn be a local
maximum of u − φ. Without loss of generality, we assume that φ(t, x) =u(t, x). We have to prove that
∂tφ(t, x) + Lφ(t, x) + f(t, x, u(t, x),∇φσ(t, x)) ≥ 0.
If not, there exist δ > 0 and 0 < α ≤ T − t such that
u(s, y) ≤ φ(s, y), ∂tφ(s, y) + Lφ(s, y) + f(s, y, u(s, y),∇φσ(s, y)) ≤ −δ
as soon as t ≤ s ≤ t+ α and |x− y| ≤ α.Consider the stopping time
τ = infs ≥ t : |Xt,xs − x| ≥ α ∧ (t+ α).
24
We consider two couples of processes. On the one hand, (Y ′s , Z′s) := (φ(s ∧
τ,Xt,xs∧τ ),1s≤τφσ(s,Xt,x
s )) solves
Y ′s = φ(τ,Xt,xτ ) +
∫ t+α
s−1r≤τ∂tφ+ Lφ(r,Xt,x
r )dr −∫ t+α
sZ ′sdWr.
On the other hand, (Y t,xs∧τ ,1s≤τZ
t,xs ) solves the BSDE
Ys = Yt+α +∫ t+α
s1r≤τf(r,Xt,x
r , Yr, Zr)dr −∫ t+α
sZsdWr.
By the Markov property, Y t,xs = u(s,Xt,x
s )
Ys = u(τ,Xt,xτ ) +
∫ t+α
s1r≤τf(r,Xt,x
r , u(r,Xt,xr ), Zr)dr −
∫ t+α
sZsdWr.
By the definition of τ , u(τ,Xt,xτ ) ≤ φ(τ,Xt,x
τ ) and
f(s,Xt,xs , u(s,Xt,x
s ),∇xφσ(s,Xt,xs ) + ∂tφ+ Lφ(s,Xt,x
s ) ≤ −δ.
By the strict version of comparison theorem, u(t, x) = Yt < Y ′t = φ(t, x)which is a contradiction.
4 Part 3: Quadratic BSDEs and Utility Maxi-mization
4.1 Problem of utility maximization
The financial market consists of one bond with interest rate zero and d ≤ mstocks. In case d < m we face an incomplete market. The price process ofstock i evolves according to the equation
dSitSit
= bitdt+ σitdBt, i = 1, . . . , d, (15)
where bi (resp. σi) is a R– valued (resp. R1×m–valued) stochastic process.Denote σ = ((σ1)∗, · · · , (σd)∗)∗ as the volatility matrix whose rank is d ( i.e.σσtr is invertible P-a.s. ) The predictable Rm–valued process ( called therisk premium ) is defined by:
θt = σtrt (σtσtrt )−1bt, t ∈ [0, T ].
25
A d–dimensional Ft–predictable process π = (πt)0≤t≤T is called tradingstrategy if
∫π dSS is well defined, e.g.
∫ T0 ‖πtσt‖
2dt <∞ P–a.s. For 1 ≤ i ≤ d,the process πit describes the amount of money invested in stock i at time t.The number of shares is πi
t
Sit.
The wealth process Xπ of a trading strategy π with initial capital xsatisfies the equation
Xπt = x+
d∑i=1
∫ t
0
πi,uSi,u
dSi,u = x+∫ t
0πuσu(dBu + θudu).
Suppose our investor has a liability F at time T .Let us recall that for α > 0 the exponential utility function is defined as
U(x) = − exp(−αx), x ∈ R.
We allow constraints on the trading strategies. Formally, they are sup-posed to take values in a closed set, i.e. πt(ω) ∈ C, with C ⊆ R1×d, and0 ∈ C.
Definition 37 [Admissible strategies with constraints ] Let C be aclosed set in R1×d and 0 ∈ C. The set of admissible trading strategies ADconsists of all d–dimensional predictable processes π = (πt)0≤t≤T which sat-isfy
∫ T0 |πtσt|
2dt <∞ and πt ∈ C P-a.s., as well as
exp(−αXπτ ) : τ stopping time with values in [0, T ]
is a uniformly integrable family.
The investor wants to solve the maximization problem
V (x) := supπ∈AD
E
[− exp
(−α
(x+
∫ T
0πtdStSt− F
))], (16)
where x is the initial wealth. V is called value function.This problem has been studied by many authors, but they suppose that
the constraint is convex in order to apply convex duality.
4.2 Martingale method
In order to find the value function and an optimal strategy one constructsa family of stochastic processes R(π) with the following properties:
26
• R(π)T = − exp(−α(Xπ
T − F )) for all π ∈ AD;
• R(π)0 = R0 is constant for all π ∈ AD;
• R(π) is a supermartingale for all π ∈ AD and there exists a π∗ ∈ ADsuch that R(π∗) is a martingale.
The process R(π) and its initial value R0 depend of course on the initialcapital x.
Given processes possessing these properties we can compare the expectedutilities of the strategies π ∈ AD and π∗ ∈ AD by
E[− exp(−α(XπT − F ))] ≤ R0(x) = E[− exp(−α(Xπ∗
T − F ))] = V (x), (17)
hence π∗ is the desired optimal strategy.Construction of R(π):
R(π)t := − exp(−α(X(π)
t − Yt)), t ∈ [0, T ], π ∈ AD,
where (Y,Z) is a solution of the BSDE
Yt = F −∫ T
tZsdBs +
∫ T
tf(s, Zs)ds, t ∈ [0, T ].
In these terms one is bound to choose a function f for which R(π) is asupermartingale for all π ∈ AD and there exists a π∗ ∈ AD such that R(π∗)
is a martingale. This function f also depends on the constraint set (C) where(πt) takes its values. One gets then
V (x) = R(π,x)0 = − exp(−α(x− Y0)), for all π ∈ AD.
In order to satisfy the supermartingale and the martingale properties,one finds
f(t, z) =α
2minπ∈C|πσ − (z +
1αθt)|2 − zθt −
12α|θt|2.
The function f is well defined because it only depends on the distancebetween a point and a closed set.
Important: the generator f is of quadratic growth with respectto z!
27
Lemma 38 Suppose that both the risk premium θ and the liability F arebounded. Then, the value function of the optimization problem (16) is givenby
V (x) = − exp(−α(x− Y0)),
where Y0 is defined by the unique solution (Y,Z) of the BSDE
Yt = F −∫ T
tZsdBs +
∫ T
tf(s, Zs)ds, t ∈ [0, T ], (18)
withf(t, z) =
α
2minπ∈C|πσ − (z +
1αθ)|2 − zθ − 1
2α|θ|2.
There exists an optimal trading strategy π∗ ∈ AD with
π∗t ∈ argmin|πσ − (Zt +1αθt)|, π ∈ C, t ∈ [0, T ], P − a.s. (19)
4.3 The simplest example
Consider
Yt = ξ +12
∫ T
t|Zs|2ds−
∫ T
tZsdWs (20)
Proposition 39 (20) admits a solution if and only if E[eξ] < ∞. In thiscase, the solution is given by
Yt = lnE[eξ|Ft].
Proposition 40 If E[eξ] < ∞, then (20) admits a unique solution suchthat eY is in class (D).
4.4 The bounded case
We consider the following real valued BSDE
Yt = ξ +∫ T
tf(s, Ys, Zs)ds−
∫ T
tZs · dWs, 0 ≤ t ≤ T, (21)
where the generator f : [0, T ]×R×Rd → R is continuous in (y, z). We callthe equation (21) as a Quadratic BSDE if
f(t, y, z) ≤ α+ β|y|+ 12|z|2,
where α, β, γ are nonnegative real numbers.
Theorem 41 [M. Kobylanski, 2000] If ξ is bounded, then the BSDE (21)has a bounded solution.
28
4.5 Unbounded Quadratic BSDEs
Boundedness of ξ is not necessary to construct a solution. In fact, somefinite exponential moment of ξ is enough!
Theorem 42 [Existence of Solution, Briand & Hu, 2006] Let us assumethat E
[exp
(γeβT |ξ|
)]< +∞.
Then, (21) has at least a solution such that
|Yt| ≤ αTeβT +1γ
logE(
exp(γeβT |ξ|
)| Ft). (22)
Theorem 43 (Briand and Hu, 2008) Let us suppose moreover that f isLipschitz w.r.t. y and convex in z. Then for any FT -measurable ξ withE[exp(λ|ξ|)] <∞ for any λ > 0, (21) admits a unique solution in the classE = Y. : E[eλ sup |Y |] < +∞ ∀λ > 0.
Theorem 44 (Delbaen, Hu and Richou, 2011) The uniqueness holds for(Y, Z) such that there exists p > γ,
E[epY∗] <∞.
Theorem 45 (Delbaen, Hu and richou, 2013) Assume moreover that f isstrictly convex and independent of y. Then uniqueness holds within the class(Y,Z) such that eγY
∗is in class (D).
4.6 Superquadratic BSDEs
Let us consider the following BSDE:
Yt = ξ −∫ T
tg(Zs)ds+
∫ T
tZsdBs, (23)
where g is convex with g(0) = 0, and is superquadratic, i.e.
lim supz→∞
g(z)|z|2
=∞;
and ξ is a bounded FT -measurable random variable.The goal here is to look for a solution (Y,Z) such that Y is a bounded
process.Different from BSDEs with quadratic growth, the bounded solution to
the BSDE with superquadratic growth does not always exist.
29
Theorem 46 (Non-existence) There exists η ∈ L∞(FT ) such that BSDE(23) with sup-quadratic growth has no bounded solution.
Even if the BSDE has a bounded solution, the solutions are not unique.The main reason is that the generator g is superquadratic which makes∫ t0 g(Zr)dr grow much faster that
∫ t0 ZrdBr with respect to Z. Following
this observation, we can construct other solutions.
Theorem 47 (Non-uniqueness) If the BSDE (g, ξ) with superquadraticgrowth has a bounded solution Y for some ξ ∈ L∞(FT ), then for each y < Y0,there are infinitely many bounded solutions X with X0 = y.
The BSDE with superquadratic growth is ill-posed. However, in theparticular Markovian case, solutions to BSDE exist.
Define the diffusion process Xt,x be the solution to the SDE:
Xs = x+∫ s
tb(r,Xr)dr +
∫ s
tσdBr, t ≤ s ≤ T, (24)
where b is Lipschitz with respect to x, and σ is a constant (matrix).Let us consider the BSDE (23) with ξ = Φ(Xt,x
T ):
Ys = Φ(Xt,xT )−
∫ T
sg(Zr)dr +
∫ T
sZrdBr.
Theorem 48 Let us suppose that Φ is bounded and continuous. Then thereexists a solution (Y,Z) to Markovian BSDE.
References
[1] Briand, P. and Hu, Y., BSDE with quadratic growth and unboundedterminal value. Probab. Theory Related Fields 136 (2006), no. 4,604C618.
[2] El Karoui, N., Peng, S., and Quenez, M. C., Backward stochastic dif-ferential equations in finance. Math. Finance 7 (1997), no. 1, 1C71.
[3] Kobylanski, M., Backward stochastic differential equations and partialdifferential equations with quadratic growth. Ann. Probab. 28 (2000),no. 2, 558C602.
[4] Pardoux, E. and Peng, S. G., Adapted solution of a backward stochasticdifferential equation. Systems Control Lett. 14 (1990), no. 1, 55C61.
30