On the solvability of forward-backward stochastic differential equations driven by Teugels Martingales

0.28 MB·

by  Dalila Guerdouh

On the solvability of forward-backward stochastic
differential equations driven by Teugels Martingales
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1
0
Dalila Guerdouh, Nabil Khelfallah, Brahim Mezerdi
2
University of Biskra, Laboratory of Applied Mathematics,
n
a
Po. Box 145 Biskra (07000), Algeria
J ∗

9
2

]
R
Abstract
P
. We deal with a class of fully coupled forward-backward stochastic differential equations
h
t (FBSDE for short), driven by Teugels martingales associated with some Lévy process. Un-
a
m der some assumptions on the derivatives of the coefficients, we prove the existence and
[ uniqueness of a global solution on an arbitrarily large time interval. Moreover, we estab-

lish stability and comparison theorems for the solutions of such equations. Note that the
1
v present work extends known results by Jianfeng Zhang (Discrete Contin. Dyn. Syst. Ser.
6 B 6 (2006), no. 4, 927–940), proved for FBSDEs driven by a Brownian motion, to FBSDEs
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driven by general Lévy processes.
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8 Keywords: Forward-backward stochastic differential equations; Teugels Martingale;
0 Lévy process.
.
1
0
7 1 Introduction
1
:
v
Let (L ) be a R valued Lévy process defined on a complete filtered probability space
Xi t 0≤t≤T −
Ω, ,( ) ,P satisfying the usual conditions. Assume that the Lévy measure ν(dz) corre-
r F Ft t≥0
a sponding to the Lévy process L satisfies:
t
(cid:0) (cid:1)
(i) (1 z2)ν(dz) < ,
∧ ∞
R
(ii)Rthere exist α > 0 such that for every ε > 0,
eα|z|ν(dz) < .
∞
Z
]−ε,ε[c
Assumptions (i) and (ii) imply in particular that the random variable L(t) has moments of all
orders. We also assume that = σ(L ,s t) , where denotes the σ field
t 0 s 1 2
F F ∨ ≤ ∨ N G ∨ G −
generated by and is the totality of the P-negligible sets.
1 2
G ∪G N
∗E-mail addresses: [email protected],[email protected],[email protected].
1
The aim of this work is to prove existence and uniqueness of solutions of the following coupled
forward-backward stochastic differential equation (FBSDE for short)
t ∞ t
X = X + f (s,w,X ,Y ,Z )ds+ σi(s,w,X ,Y )dHi,
t 0 s s s s− s− s
 0 i=10 (1.1)
R T PR∞ T
 Y = ϕ(X )+ g(s,w,X ,Y ,Z )ds ZidHi,
 t T s s s − s s
t i=1 t
where t [0,T], H = (Hi)∞ areRpairwise strongly orthonPormRal Teugels martingales associated
with the∈Lévy proctess L t. iF=o1r any R-valued and -measurable random vector X , satisfying
t 0 0
E X 2 < , we are looking for an R R l(R)-Fvalued solution (X ,Y ,Z ) on an arbitrarily
0 t t t
| | ∞ × ×
fixed large time duration, which is square-integrable and adapted with respect to the filtration
generated by L and satisfying
t t 0
F F
t
E X 2 + Y 2 + Z 2 dt < .
t t t
| | | | | | ∞
Z
0 (cid:0) (cid:1)
The existence and uniqueness of solutions of FBSDEs without the Teugels part have been
widely studied by many authors (see, e.g. [1], [4], [6], [7], [10], [11], and [15]). The first study
of FBSDEs has been performed by Antonelli [1] in the early 1990s. The author has used the
contraction mapping technique to obtain a local existence and uniqueness result in a small
time interval. Hu and Peng [6] have used a probabilistic method to establish an existence and
uniqueness result, under certain monotonicity conditions, in the case where the forward and
backward components have the same dimension. Then Hamadène [5] improved their result by
proving it under weaker monotonicity assumptions. Peng and Wu provided in [11] more general
results by extending the two above results, without the restriction on the dimensions of the
forward and backward parts.
In spite of the large literature devoted to the Brownian case as we have mentioned above,
there are relatively a few results on FBSDEs driven by Teugels Martingales. To the best of our
knowledge, the first paper dealing with this kind of equations driven by Lévy processes is [12],
where the authors have proved the existence and uniqueness via the solution of its associated
partial integro-differential equation (PIDE for short). Then Baghery et al. [2] proved under some
monotonicity assumptions, the existence and uniqueness of solutions on an arbitrarily fixed large
time duration.
Motivated by the above results and by imposing an assumption on the derivatives of the
coefficients, introduced by Zhang [16], we establish two main results. We shall first prove the
existence and uniqueness of the solution of the FBSDE 1.1, without any restriction on the time
duration. The main idea of the proof is to construct the solution on small intervals, and then
extend it piece by piece to thewhole interval. Ina second step, we prove stability andcomparison
theorems for the solutions. Let us point out that our work extends the results of Jianfeng
Zhang (Discrete Contin. Dyn. Syst. Ser. B 6 (2006), no. 4, 927–940), to FBSDEs driven by
general Lévy processes. We note that much of the technical difficulties coming from the Teugels
martingales are due to the fact that the quadratic variation [Hi,Hj] is not absolutely continuous,
with respect to the Lebesgue measure. To overcome these difficulties, we use the fact that the
predictable quadratic variation process Hi,Hj is equal to δ t and that [Hi,Hj] Hi,Hj
h it ij t −h it
is a martingale.
2
This paper is organized as follows. In Section 2, we give some preliminaries and notations
about Teugels martingales. In Section 3, we give some assumptions and provide our main results.
The proofs are provided in the last section.
2 Notations and assumptions
Let us recall briefly the L2 theory of Lévy processes as it is investigated in Nualart-Schoutens [8].
Aconvenient basis formartingalerepresentation is provided by theso-called Teugels martingales.
This means that this family has the predictable representation property.
Denote by ∆L = L L where
t t − t−
L = lim L , t > 0,
t− s
s→t,s<t
and define the power jump processes by
L if i = 1;
t
L(ti) = ( Ls)i if i 2.
( △ ≥
0<s≤t
P
If we denote
Y(i) = L(i) E L(i) ,i 1,
t t − t ≥
h i
with
E L(1) = E[L ] = tE[L ] = tm ,
t t 1 1
h i
and, for i 2
≥
∞
E L(i) = E ( L(s))i = t ziν(dz) = tm .
t △ i
h i "0X<s≤t # Z−∞
Then the family of Teugels martingales (Hi)∞ , is defined by
t i=1
j=i
Hi = a Y(j).
t ij t
j=1
X
The coefficients a correspond to the orthonormalization of the polynomials 1, x, x2, ...
ij
with respect to the measure µ(dx) = x2ν(dx) + δ (dx). Then (Hi)∞ is a family of strongly
0 t i=1
orthogonal martingales such that Hi,Hj = δ .t and [Hi,Hj] Hi,Hj is a martingale, see
h it ij −h it
[8, 13].
The following lemma which gives some useful properties of the Teugels martingale will be
needed in the sequel.
Lemma 2.1. i) The process Hi can be represented as follows:
t
Hi = q (0)B + p (x)N˜ (t,dx)
t i−1 t i
R
Z
where B be a Brownian motion, and N˜ (t,dx) is the compensated Poisson random measure that
t
corresponds to the pure jump part of L and the polynomials q (0) and p (x) associated to L .
t i−1 i t
3
ii) The polynomials p and q are linked by the relation:
i j
p (x)p (x)v(dx) = δ q (0)q (0).
i j ij i−1 j−1
−
R
Z
Proof. See [12]. (cid:3)
In the rest of this section, we list all the notations that will be frequently used throughout
this work.
l2 : the Hilbert space of real-valued sequences x = (x ) with norm
n n≥0
1
∞ 2
x = x < .
i
k k ∞
!
i=1
X
Let us define
l2(R) : the space of R-valued process fi such that
{ }i≥0
1
∞ 2
fi 2 < .
R ∞
!
i=1
X(cid:13) (cid:13)
(cid:13) (cid:13)
l2 (0,T,R) : the Banach space of l2(R) valued predictable processes such that
F − Ft−
1
T ∞ 2
E fi(t) 2 < .
R ∞
Z0 i=1 !
X(cid:13) (cid:13)
(cid:13) (cid:13)
2 (0,T,R) : the Banach space of R valued adapted and càdlàg processes such that
SF − Ft−
1
2
E sup f (t) 2 < .
| | ∞
(cid:18) 0≤t≤T (cid:19)
L2(Ω, ,P,R) : the Banach space of R valued, square integrable random variables on
F −
(Ω, ,P). Here and in what follows, for notational simplicity, we shall denote
F
t T
σ(s,w,X ,Y )dH and Z dH
0 s− s− s t s s
instead of R R
∞ ∞
tσi(s,w,X ,Y )dHi and T ZidHi
0 s− s− s t s s
i=1 i=1
respectively, where Z =PZRi ∞ , σ = σi ∞ , σi : [0,T] PΩ RR R l2(R). Further, for the
s { s}i=1 s { s}i=1 × × × →
notational simplicity, we have suppressed w and we will do so below. We also use the following
notation
M2(0,T) = 2 (0,T,R) 2 (0,T,R) l2 (0,T,R).
SF ×SF × F
The following assumptions will be considered in this paper.
4
We suppose that the coefficients
f : [0,T] Ω R R l2(R) R,
σ : [0,T]×Ω×R×R× l2(R)→,
g : [0,T] ×Ω ×R ×R →l2(R) R,
× × × × →
ϕ : Ω R R,
× →
are progressively measurable, such that:
(H ) There exist λ, λ > 0, such that t [0,T], (x,y,z) and (x′,y′,z′) in R R l(R)
1 0
∀ ∈ ∀ × ×
f (t,x,y,z) f (t,x′,y′,z′) λ x x′ + y y′ + z z′ ,
| − | ≤ | − | | − | k − kl2(R)
(cid:16) (cid:17)
σ(t,x,y) σ(t,x′,y′) 2 λ2 x x′ 2 + y y′ 2 ,
| − | ≤ | − | | − |
(cid:16) (cid:17)
g(t,x,y,z) g(t,x′,y′,z′) λ x x′ + y y′ + z z′ ,
| − | ≤ | − | | − | k − kl2(R)
ϕ(x) ϕ(x′) λ(cid:16)( x x′ ). (cid:17)
0
| − | ≤ | − |
(H2) The functions f,g,σ,ϕ are differentiable with respect to x, y, z with uniformly bounded
derivatives such that
σ f = 0 and f +σ f +σ g = 0. (2.1)
y z y x z y z
Let us mention that assumption (H ) has been introduced bfor the first time by Zhang [16]
2
in the case of FBSDEs without jumps.
3 The main results
3.1 Existence and uniqueness
The following theorem gives the existence of a solution in a small time duration.
Theorem 3.1. Suppose that (H1) is satisfied. Assume further that
T
V2 =△ E X 2 + ϕ(0) 2 + f (t,0,0,0) 2 + σ(t,0,0) 2 + g(t,0,0,0) 2 dt < .
0 | 0| | | | | k kl2(R) | | ∞
(cid:26) Z0 (cid:27)
h i
Then, for every -measurable random vector X , there exists a constant δ depending only on λ
0 0
F
and λ , such that for T δ, equation (1.1) has a unique solution which belongs to M2(0,T).
0
≤
The following proposition gives a priori estimates, which shows in particular the continuous
dependence of the solution upon the data.
Proposition 3.1. Under the same assumptions of the Theorem 3.1, there exist δ and C de-
0
pending on λ and λ , such that for T δ, the following estimates hold true:
0
≤
i)
1
T 2
Π = E sup X 2 + Y 2 + Z 2 dt C V .
k k | t| | t| k tkl2(R) ≤ 0 0
(cid:18)0≤t≤T Z0 (cid:19)
(cid:2) (cid:3)
5
ii)
T p
E sup X 2p + Y 2p + Z 2 dt <
| t| | t| k tkl2(R) ∞
(cid:26)0≤t≤T (cid:18)Z0 (cid:19) (cid:27)
(cid:2) (cid:3)
The next Theorem extends the result in Theorem 3.1 to arbitrary large time duration.
Theorem 3.2. Assume (H1), (H2) and V2 < . Then:
0 ∞
i) Equation (1.1) has a unique solution Π M2(0,T).
∈
ii) The following estimate holds
Π 2 CV2.
k k ≤ 0
3.2 Stability theorem
The following results state the stability of the solution of FBSDE (1.1) with respect to the initial
condition and the data. This means that the solution of equation (1.1) does not change too much
under small perturbations of the data. In other words, the trajectories which are close to each
other at specific instant should therefore remain close to each other at all subsequent instants.
To state the next theorem and its corollary, let us consider Πi,i = 0,1 the solutions of (1.1)
corresponding to (fi,σi,gi,ϕi). We shall consider the following notations, ∆Π =∆ Π1 Π0 and
−
for any function h =∆ f,σ,g,ϕ, we set ∆h =∆ h1 h0.
−
Theorem 3.3. Assume that (fi,σi,gi,ϕi,Xi),i = 0,1, satisfy the same conditions of Theorem
0
3.2. Then
T
∆Π 2 CE ∆X 2 + ∆ϕ X1 2 + ∆f 2 + ∆σ 2 + ∆g 2 t,Π1 dt .
k k ≤ | 0| T | | k kl2(R) | | t
(cid:26) Z0 h i (cid:27)
(cid:12) (cid:0) (cid:1)(cid:12) (cid:0) (cid:1)
Corollary 3.1. Suppose that ((cid:12)fn,σn,ϕn,(cid:12)gn,Xn), for n = 0,1... satisfy the same conditions of
0
Theorem 3.2. Moreover assume that:
i) Xn X0 in L2.
0 → 0
ii) for h =∆ f,σ,ϕ,g , hn(t,Π) h0(t,Π) as n .
→ → ∞
iii) E Xn X0 2 + ϕn ϕ0 2(0)+ T fn f0 2 + σn σ0 2 + gn g0 2 (t,0,0,0)dt
| 0 − 0| | − | 0 | − | k − kl2(R) | − | →
0 n h i o
R
Then if Πn (resp.Π) denotes the solution of (1.1) corresponding to (fn,σn,ϕn,gn,Xn) (resp.
0
(f,σ,ϕ,g,X0), we obtain
0
Πn Π0 0 as n + .
− → −→ ∞
(cid:13) (cid:13)
3.3 Comparison theore(cid:13)m (cid:13)
In what follows we provide, under the same assumptions as for the existence and uniqueness
results, another important result, which is the comparison theorem. Let (X,Y,Z) be the solution
to the following LFBSDE:
X = t(a1X +b1Y +c1Z )ds+ t(a2X +b2Y )dH ,
t 0 s s s s s s 0 s s s s s (3.1)
Y = PX +α+ T (a3X +b3Y +c3Z +β )ds T Z dH .
( t R T t s s s s R s s s − t s s
Then we have the following propositRion, which is the linear version of tRhe next theorem.
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Proposition 3.2. Assume ai , bi , ci λ, P λ and (H ) holds true. Assume further that
| t| | t| | t| ≤ | | ≤ 0 2
α 0 and β 0. Then
≥ s ≥
Y 0.
0
≥
Further we have the following general result. Let Πi,i = 0,1, be the solution of the following
FBSDE:
Xi = X + tf (s,Πi)ds+ tσ s,Xi ,Yi dH ,
t 0 0 s 0 s− s− s (3.2)
Yi = ϕi(Xi)+ T gi(s,Πi)ds T ZidH ,i = 0,1
( t TR t s R −(cid:0) t s s (cid:1)
Theorem 3.4. Let Πi,i = 0,1, be the soRlutions of the FBRSDEs (1.1). If
i) (f,σ,gi,ϕi),i = 0,1 satisfy (H ) and V2 < .
2 0 ∞
ii) For any (t,Π),ϕ0(X) ϕ1(X) and g0(t,Π) g1(t,Π). Then
≤ ≤
Y0 Y1.
0 ≤ 0
We would like to mention that the above comparison theorem holds true only at time t = 0.
We cannot get the result in the whole interval [0,T], even in the Brownian case. See for instance,
the counterexample which is given in [14].
Remark 3.1. We should point out that the following cases are in fact, involved in our present
study.
1. FBSDEs driven by Brownian motion: If ν = 0, then all non–zero degree polynomials
q (x) will vanish, H(1) = W is a standard Brownian motion and H(i) = 0, for i 2.
i−1 t t t ≥
2. FBSDEs driven by Poisson Process: assume that µ only has mass at 1, then H(i) =
t
N λt is the compensated Poisson process with intensity λ and also H(i) = 0, for i 2.
t − t ≥
For example, If we have ν(dx) = ∞ α δ (dx), where δ (dx) denotes the positive mass
j=1 j βj βj
measure at β R of size 1. Then, The process L takes the form
j ∈ P ·
∞
L = at+ N(j) +α t ,
t t j
Xj=1 (cid:16) (cid:17)
where N(j) +∞ denotethe sequenceof independentPoissonprocesswith parameters α +∞.
{ t }j=1 { j}j=1
In this case
∞
β
H(1) = 1 N(j) +α t
t √α t j
j
Xj=1 (cid:16) (cid:17)
4 Proofs and technical results
4.1 Small time duration
In this subsection, we shall start by giving and proving the following technical Lemma, which
will be used in the proof of Theorem 3.1. Let us introduce the following decoupled FBSDE:
X˜ = X + tf s,X˜ ,Y ,Z ds+ tσ s,X˜ ,Y dH ,
t 0 0 s s s 0 s− s− s
(4.1)
 Y˜ = ϕ(X R)+ (cid:16)T g s,X˜ ,Y˜(cid:17),Z˜ dRs (cid:16)T Z˜ dH . (cid:17)
 t T t s s s − t s s
(cid:16) (cid:17)
R R

7
Lemma 4.1. Assume that all the conditions in Theorem (3.1) are satisfied. Let X˜ ,Y˜ ,Z˜ and
s s s
U˜ ,V˜,W˜ belong to M2(0,T) and satisfy the equation (4.1), then there exists(cid:16)three cons(cid:17)tants
t t s
c(cid:16),c′ and c′′(cid:17)depending on λ and λ , such that the following estimates hold true
0
T
1 cT12 E sup X˜ U˜ 2 cT12 E sup Y V 2 +E Z W 2 ds , (4.2)
− t − t ≤  | s − s| k s − skl2(R) 
0≤t≤T 0≤s≤T
(cid:16) (cid:17) (cid:12) (cid:12) Z0
(cid:12) (cid:12)
 
(cid:12) (cid:12)
2 2
(1 c′′T)E sup Y˜ V˜ c′′(1+T)E sup X˜ U˜ , (4.3)
t t s s
− − ≤ −
(cid:18)0≤t≤T (cid:19) (cid:18)0≤s≤T (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
E T Z˜ W˜ 2 ds c′ (cid:12)(cid:12)(1+T(cid:12)(cid:12))E sup X˜ U˜ 2 +T(cid:12)(cid:12) E sup(cid:12)(cid:12) Y˜ V˜ 2 .
s s s s s s
(cid:20)Z0 (cid:13) − (cid:13)l2(R) (cid:21) ≤ (cid:18) (cid:18)0≤s≤T (cid:12) − (cid:12) (cid:19) (cid:18)0≤s≤T (cid:12) − (cid:12) (cid:19)(cid:19)
(4.4)
(cid:13) (cid:13) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:13) (cid:13) (cid:12) (cid:12) (cid:12) (cid:12)
Proof of Lemma 4.1. Let us consider (X ,Y ,Z ) , X˜ ,Y˜,Z˜ ,
t t t 0≤t≤T t t t
0≤t≤T
(U ,V ,W ) , U˜ ,V˜,W˜ M2(0,T). First,(cid:16)we procee(cid:17)d to prove (4.2). Applying
t t t 0≤t≤T t t t 0≤t≤T ∈
(cid:16) 2 (cid:17)
Itô’s formula to X˜ U˜ , taking expectation and using the fact that [Hi,Hj] Hi,Hj is
t − t t −h it
an -martingale(cid:12)and H(cid:12)i,Hj = δ t, then there exists a constant c, depending on λ such that
Ft (cid:12) h (cid:12) it ij
(cid:12) (cid:12)
2
E sup X˜ U˜ c E T X˜ U˜ X˜ U˜ + Y V + Z W 2 ds
t − t ≤ 0 s − s s − s | s − s| k s − skl2(R)
0≤t≤T
(cid:12) (cid:12) h (cid:12) (cid:12)(cid:16)(cid:12) (cid:12) (cid:17)
+E T (cid:12) X˜ U˜(cid:12) 2 + Y R V(cid:12) 2 ds (cid:12) (cid:12) (cid:12)
0 (cid:12) s − (cid:12)s | s − (cid:12)s| (cid:12) (cid:12) (cid:12)
(cid:18)(cid:12) (cid:12) (cid:19) (cid:21)
+2ER sup(cid:12) t X˜ (cid:12) U˜ σ s,X˜ ,Y σ s,U˜ ,V dH .
(cid:12) 0 s(cid:12)− − s− s− s− − s− s− s
0≤t≤T
(cid:12) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:12)
(cid:12)R (cid:12)
Burkholder-Davis(cid:12)-Gundy’s inequality applied to the martingale (cid:12)
t
X˜ U˜ σ s,X˜ ,Y σ s,U˜ ,V dH
s s s− s− s− s− s
− −
Z0
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
yields the existence of a constant C > 0, such that
E sup t X˜ U˜ σ s,X˜ ,Y σ s,U˜ ,V dH
0 s − s s− s− − s− s− s
0≤t≤T
(cid:12) (cid:16) (cid:17)(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) (cid:12) 1
CE ·(cid:12)RX˜ U˜ σ s,X˜ ,Y σ s,U˜ ,V dH (cid:12) 2
≤ 0(cid:12) s− − s− s− s− − s− s− s(cid:12)
(cid:16)h (cid:16) (cid:17)(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) i(cid:17)
R
Moreover, since Hi,Hi = δ t and [M] = M + ψ , where ψ is a uniformly integrable
h i ij t h it t t
8
martingale starting at 0, then
1/2
2
E sup t X˜ U˜ σ s,X˜ ,Y σ s,U˜ ,V dH
0 s − s s− s− − s− s− s
(cid:18)0≤t≤T (cid:19)
CE (cid:12)(cid:12)R· X(cid:16)˜ U˜ (cid:17)σ(cid:16) s(cid:16),X˜ ,Y (cid:17) σ s(cid:16),U˜ ,V (cid:17)(cid:17)dH (cid:12)(cid:12)1/2
(cid:12)0 s − s s− s− − s− s− s (cid:12)
(cid:16)h· (cid:16) (cid:17)(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17) i(cid:17) 1/2
= CE R X˜ U˜ σ s,X˜ ,Y σ s,U˜ ,V dH +ψ
s− − s− s− s− − s− s− s t
(cid:18)(cid:28)0 (cid:29) (cid:19)
R (cid:16) T (cid:17)2 (cid:16) (cid:17) (cid:16) 2(cid:17) 1/2
= CE X˜ U˜ σ s,X˜ ,Y σ s,U˜ ,V ds .
s s s s s s
(cid:18)0 − − l2(R) (cid:19)
(cid:12) (cid:12) (cid:13) (cid:16) (cid:17) (cid:16) (cid:17)(cid:13)
R
(cid:12) (cid:12) (cid:13) (cid:13)
Then, modifying c if neces(cid:12)sary, we (cid:12)ha(cid:13)ve (cid:13)
2
E sup X˜ U˜
t t
−
(cid:20)0≤t≤T (cid:21)
(cid:12) (cid:12)
2
cT1/2 E sup X˜ U˜ +E sup(cid:12) Y V(cid:12) 2 +E T Z W 2 ds ;
≤ s − s (cid:12) | s − (cid:12)s| 0 k s − skl2(R)
(cid:18) (cid:18)0≤s≤T (cid:19) (cid:18)0≤s≤T (cid:19) (cid:19)
(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12) R
which implies that, (cid:12) (cid:12)
2 T
1 cT1/2 E sup X˜ U˜ cT1/2 E sup Y V 2 +E Z W 2 ds .
− t − t ≤ | s − s| k s − skl2(R)
(cid:18)0≤t≤T (cid:12) (cid:12) (cid:19) (cid:18) (cid:18)0≤s≤T (cid:19) (cid:18)Z0 (cid:19)(cid:19)
(cid:0) (cid:1)
(cid:12) (cid:12)
On the other hand, b(cid:12)y applyin(cid:12)g Itô’s formula to Y˜ V˜ 2, we get
t t
−
(cid:12) (cid:12)
2 T 2 (cid:12) (cid:12)
Y˜ V˜ + Z˜ W˜ ds (cid:12) (cid:12)
t t s s
− t − l2(R)
(cid:12) (cid:12) (cid:13) (cid:13)
(cid:12)(cid:12)= ϕ X˜(cid:12)(cid:12)T R ϕ(cid:13)(cid:13) U˜T 2 +(cid:13)(cid:13) 2 T Y˜s V˜s g s,X˜s,Y˜s,Z¯s g s,U˜s,V˜s,W˜s ds (4.5)
− − −
t
(cid:12)T(cid:16) (cid:17) (cid:16) (cid:17)(cid:12) R (cid:16) (cid:17)T(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
−2(cid:12)(cid:12) Y˜s −V˜s Z˜s −(cid:12)(cid:12)W˜s dHs − i,j Z˜si −W˜si Z˜sj −W˜sj d[Hi,Hj]s.
t t
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17)
R P R
Thus, bytakingexpectations, invoking theassumption(H )andusing thefactthat Z˜i W˜ i
1 s − s −
Hi,Hj is an -martingale and Hi,Hj = δ t, one can show that there exists a(cid:16)constant(cid:17)c′,
h it Ft h it ij
depending on λ and λ , such that
0
2 2
E T Z˜ W˜ ds c′ E X˜ U˜
0 s − s l2(R) ≤ T − T
(cid:20)
+RE (cid:13)(cid:13)T Y˜ V˜(cid:13)(cid:13) X˜ U˜ + (cid:12)(cid:12)Y˜ V˜ +(cid:12)(cid:12) Z˜ W˜ 2 ds .
0(cid:13) s − s(cid:13) s − s (cid:12) s − s (cid:12) s − s l2(R)
(cid:18) (cid:19) (cid:21)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:13) (cid:13)
R
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:13) (cid:13)
Using the fact that ab (cid:12) 1 a(cid:12)2 +(cid:12) b 2 for(cid:12)any(cid:12) a,b R(cid:12) , w(cid:13)e have (cid:13)
| | ≤ 2 | | | | ∈
(cid:0) 2(cid:1) 2
E T Z˜ W˜ ds c′ (1+T)E sup X˜ U˜
0 s − s ≤ s − s
(cid:20) 0≤s≤T
R (cid:12) (cid:12) 2 1 (cid:12) 2 (cid:12)
+TE(cid:12) sup Y˜(cid:12) V˜ + E T Z˜ W˜(cid:12) ds(cid:12).
(cid:12)0≤s≤T s(cid:12)− s (cid:21) 2 0 s − (cid:12)s l2(R) (cid:12)
(cid:12) (cid:12) (cid:13) (cid:13)
R
(cid:12) (cid:12) (cid:13) (cid:13)
(cid:12) (cid:12) (cid:13) (cid:13)
9
By modifying c′ if necessary, we obtain
2
E T Z˜ W˜ ds
0 s − s l2(R)
(4.6)
Rc′ (cid:13)(cid:13)(1+T)E(cid:13)(cid:13)sup X˜ U˜ 2 +TE sup Y˜ V˜ 2 .
(cid:13) (cid:13) s s s s
≤ − −
(cid:20) 0≤s≤T 0≤s≤T (cid:21)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Using equality (4.5) once again, and the B(cid:12)urkholde(cid:12)r-Davis-Gund(cid:12)y inequa(cid:12)lity, we show that there
exists a constant c′′, only depending on λ and λ , such that
0
1/2
2 2 2 2
E sup Y˜ V˜ c′′ E X˜ U˜ +E T Y˜ V˜ Z˜ W˜ ds
0≤t≤T t − t ≤ " T − T (cid:18) 0 s − s s − s l2(R) (cid:19)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:13) (cid:13)
(cid:12) (cid:12) (cid:12) (cid:12) R (cid:12) (cid:12) (cid:13) 2 (cid:13)
(cid:12)+E T (cid:12)Y˜ V˜ (cid:12)X˜ U˜ (cid:12)+ Y˜ V˜ (cid:12)+ Z˜ (cid:12) W(cid:13)˜ (cid:13)ds .
0 s − s s − s s − s s − s l2(R)
(cid:18) (cid:19) (cid:21)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:13) (cid:13)
R
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:13) (cid:13)
Then, Taking into acco(cid:12)unt (4.6(cid:12)), (cid:12)using Yo(cid:12)ung(cid:12)’s inequa(cid:12)lity(cid:13)one mor(cid:13)e time, and modifying c′′ if
necessary, we get
2 2 2
E sup Y˜ V˜ c′′ (1+T)E sup X˜ U˜ +TE sup Y˜ V˜
t t s s s s
− ≤ − −
(cid:18)0≤t≤T (cid:19) (cid:20) (cid:18)0≤s≤T (cid:19) (cid:18)0≤s≤T (cid:19)(cid:21)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
2
+1E su(cid:12)p Y˜ (cid:12)V˜ (cid:12) (cid:12) (cid:12) (cid:12)
2 (cid:12) t −(cid:12) t (cid:12) (cid:12) (cid:12) (cid:12)
(cid:18)0≤t≤T (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
Then, modifying c′(cid:12)′ if nece(cid:12)ssary, we have
1 c′′T E sup Y˜ V˜ 2 c′′(1+T)E sup X˜ U˜ 2 .
t t s s
− − ≤ −
(cid:18)0≤t≤T (cid:19) (cid:18)0≤s≤T (cid:19)
(cid:16) (cid:17) (cid:12) (cid:12) (cid:12) (cid:12)
Lemma 4.1 is proved. (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:3)
Proof of Theorem 3.1. Let (X ,Y ,Z ) be a possible solution of FBSDE (1.1) and
t t t 0≤t≤T
X˜,Y˜,Z˜ be defined as in Lemma 4.1. It is clear that the process X˜ is a solution of a Forward
(cid:16)componen(cid:17)t of the SDE (4.1), whereas the couple X˜,Y˜ is a solution of a Backward component
of the SDE (4.1) SDE. Then X˜,Y˜,Z˜ is a solut(cid:16)ion of(cid:17)the above decoupled Forward Backward
SDE (4.1). To prove the exis(cid:16)tence and(cid:17)the uniqueness of the solution in M2(0,T), we use the
fixed point method. Let us define a mapping Ψ from M2(0,T) into itself defined by
Ψ(X,Y,Z) = X˜,Y˜,Z˜ .
(cid:16) (cid:17)
We want to prove that there exists a constant δ > 0, only depending on λ and λ , such that for
0
T δ, Ψ is a contraction on M2(0,T) equipped with the norm
≤
Ψ(X,Y,Z) 2 = E sup X 2 + Y 2 + T Z 2 dt .
k kM2(0,T) | t| | t| 0 k tkl2(R)
(cid:26)0≤t≤T (cid:27)
(cid:2) (cid:3) R
In order to achieve this goal, we firstly assume that T 1. Further, we set
≤
Ψ(X,Y,Z) = X˜,Y˜,Z˜ , Ψ(U,V,W) = U˜,V˜,W˜ .
(cid:16) (cid:17) (cid:16) (cid:17)
10