2.1 Itˆ o’s formula

BSDE ASSOCIATED WITH L´EVY PROCESSES AND APPLICATION TO PDIE

K. BAHLALI UFR Sciences

UVT, BP 132, 83957 La Garde Cedex, France¹ CPT, CNRS, Luminy. Case 907 13288 Marseille Cedex 9, France

E-mail: [email protected]

M. EDDAHBI Universit´e Cadi Ayyad

Facult´e des Sciences et Techniques² D´epartment de Math & Info., BP 549

Marrakech, Maroc

E-mail: [email protected]

E. ESSAKY Universit´e Cadi Ayyad

Facult´e des Sciences Semlalia³ D´epartement de Math´ematiques, BP 2390

Marrakech, Maroc E-mail: [email protected]

(Received April,2002; Revised November,2002)

We deal with backward stochastic differential equations (BSDE for short) driven by Teugel’s martingales and an independent Brownian motion. We study the existence, uniqueness and comparison of solutions for these equations under a Lipschitz as well as a locally Lipschitz conditions on the coefficient. In the locally Lipschitz case, we prove that if the Lipschitz constant LN behaves as p

log(N) in the ball B(0, N), then the correspondingBSDE has a unique solution which depends continuously on the on the coefficient and the terminal data. This is done with an unbounded terminal data. As application, we give a probabilistic interpretation for a large class of partial differential integral equations (PDIE for short).

Keywords. Backward Stochastic Differential Equations, L´evy Processes, Teugel’s Mar- tingales, Partial Differential Integral Equations, Clark-Ocone Formula.

AMS (MOS) subject classification: 60H10, 60H15

1Supported by CMEP, 077/2001.

2Supported by CNRST Maroc/CNRS France, 8310-2000 and TWAS grant 98-199 RG/MATHS/AF/AC.

3Supported by CMIFM, A.I. n^oMA/01/02.

1

1 Introduction

Since the paper [8] of Pardoux and Peng,several works have been devoted to the study of BSDEs as well as to their applications. This is due to the connections of BSDEs with stochastic optimal control and stochastic games (Hamad`ene and Lepeltier [3]) as well as to mathematical finance (El Karoui et al. [4]). Backward stochastic differential equations also appear as a powerful tool in partial differential equations where they provide probabilistic formulas for their solutions (Peng [10],Pardoux and Peng [9]).

A solution of a classical BSDE is a pair of adapted processes (Y, Z) satisfying:

Y_t=ξ+ _T

t

f(s, Y_s, Z_s)ds− _T

t

Z_sdW_s. (1.1)

When the coefficientf is uniformly Lipschitz,the BSDE (1.1) has a unique solution.

The proof is mainly based on the Itˆo martingale representation theorem.

In Nualart and Schoutens [6],a martingale representation theorem associated to L´evy processes was proved. It then is natural to extend equations (1.1) to BSDE’s driven by a L´evy process (Nualart and Schoutens [7]). In their paper [7],the authors proved the existence and uniqueness of solutions,under Lipschitz conditions on the coefficient.

In this paper,we deal with BSDE driven by both a standard Brownian motion and an independent L´evy process and having a Lipschitz,or more generally,a locally Lipschitz coefficient. In the locally Lipschitz case,we prove that if the Lipschitz constant L_N behaves as

log(N) in the ballB(0, N),then the corresponding BSDE has a unique solution. We don’t impose any boundedness condition on the terminal data. It will be assumed square integrable only. Moreover,a comparison theorem as well as a stability of solutions are established in this setting. Our results extend in particular those of ([1],[2]) to BSDE driven by a L´evy process. As an application,we give a probabilistic interpretation for a large class of partial differential integral equations.

The paper is organized as follows. In Section 2,we introduce some notations and assumptions. Section 3 is devoted to the proof of existence,uniqueness and comparison results for BSDE driven by a L´evy process,under Lipschitz conditions. Those equations are also discussed under locally Lipschitz conditions in Section 4. In Section 5,we include an application to PDIE.

2 Preliminaries and Notations

Let (Ω,F,P,F_t, W_t, L_t : t ∈ [0, T]) be a complete Wiener–L´evy space in R×R\{0}, with L´evy measureν,i.e. (Ω,F,P) is a complete probability space,{F_t:t∈[0, T]} is a right–continuous increasing family of complete subσ–algebras ofF,{Wt:t∈[0, T]}

is a standard Wiener process inRwith respect to{Ft:t∈[0, T]}and {Lt:t∈[0, T]}

is a R–valued L´evy process of the form L_t =bt+_t independent of {Wt :t ∈[0, T]}, corresponding to a standard L´evy measureν satisfying the following conditions : i)

R(1∧y²)ν(dy)<∞, ii)

]−ε,ε[^ce^λ|y|ν(dy)<∞,for everyε >0 and for someλ >0.

We assume that

Ft=σ(L_s, s≤t)∨σ(W_s, s≤t)∨ N

where N denotes the totality of P–null sets andG₁∨ G₂ denotes the σ–field generated byG₁∪ G₂.

LetH²denote the space of real valued,square integrable andFt–progressively measur- able processesφ={φt:t∈[0, T]}such that

φ²=E _T

0 |φt|²dt < ∞.

and denote byP²the subspace ofH² formed by the predictable processes.

Letl²be the space of real valued sequences (x_n)_n≥0such that_∞

i=0x²_i is finite. We shall denote byH²(l²) andP²(l²) the corresponding spaces ofl²–valued processes equipped with the norm

φ²= ∞ i=0

E _T

0 |φ⁽ⁱ⁾_t |²dt.

Let us define:

(A.1) a terminal valueξ∈L²(Ω,FT,P).

(A.2) a processf,which is a mapf : [0, T]×Ω×R×R×l²−→R,such that (i) f is progressively measurable alsof(.,0,0,0)∈ H².

(ii) There existsL >0 such that

|f(t, ω, y, u, z)−f(t, ω, y, u, z)| ≤L(|y−y|+|u−u|+z−z).

We recall the Itˆo formula for c`adl`ag semimartingales.

2.1 Itˆ o’s formula

LetX ={X_t:t∈[0, T]} be a c`adl`ag semimartingale,with quadratic variation denoted by [X] ={[X]_t:t∈[0, T]} and letF be aC² real valued function. ThenF(X) is also a semimartingale and the following formula holds:

F(X_t) = F(X₀) + _t

0 F(X_s−)dX_s+1 2

_T

0 F(X_s)d[X]^c_s (2.1)

+

0<s≤t

{F(X_s)−F(X_s−)−F(X_s−)∆X_s}.

where [X]^c(sometimes denoted byX) is the continuous part of the quadratic variation [X]. We also note that in the case whereF(x) =x²,the formula (2.1) takes the form

X_t²=X₀²+ _t

0 2X_s−dX_s+ _t

0 d[X]_s. (2.2)

Moreover ifX andY are two c`adl`ag semimartingales then we have X_tY_t=X₀Y₀+

_t

0 X_s−dY_s+ _t

0 Y_s−dX_s+ _t

0 d[X, Y]_s. (2.3) where [X, Y] stands for the quadratic covariation ofX , Y also called the bracket process.

For a complete survey in this topic we refer to Protter [11].

2.2 Predictable representation

We denote by (H⁽ⁱ⁾)_i≥1 the Teugel’s Martingales associated with the L´evy process {L_t:t∈[0, T]}. More precisely

H_t⁽ⁱ⁾=c_i,iY_t⁽ⁱ⁾+c_i,i−1Y_t⁽ⁱ⁻¹⁾+. . .+c_i,1Y_t⁽¹⁾,

where Y_t⁽ⁱ⁾ = L⁽ⁱ⁾_t −E[L⁽ⁱ⁾_t ] = L⁽ⁱ⁾_t −tE[L⁽ⁱ⁾₁ ] for all i ≥ 1 and L⁽ⁱ⁾_t are power–jump processes. That is, L⁽¹⁾_t =L_t and L⁽ⁱ⁾_t =

0<s≤t(∆L_t)ⁱ for i ≥ 2. It was shown in Nualart and Schoutens [6] that the coefficientsc_i,kcorrespond to the orthonormalization of the polynomials 1, x, x², ...with respect to the measureµ(dx) =x²ν(dx) +σ²δ₀(dx):

q_i−1=c_i,ixⁱ⁻¹+c_i,i−1xⁱ⁻²+...+c_i,1. We set

p_i(x) =xq_i−1(x) =c_i,ixⁱ+c_i,i−1xⁱ⁻¹+...+c_i,1x.

The martingales (H⁽ⁱ⁾)_i≥1 can be chosen to be pairwise strongly orthonormal mar- tingales. More details,in this subject,can be found in Nualart and Schoutens [6].

The main tool in the theory of BSDEs is the martingale representation theorem, which is well known for martingales which are adapted to the filtration of the Brownian motion or that of Poisson point process (e.g Situ [13]) or that of a Poisson random measure (e.g Ouknine [12]). A more general and interesting martingale representation theorem (proven by different ways) appeared recently in Løkka [5] and in Nualart and Schoutens [7].

Proposition 2.1: Let {M_t :t∈ [0, T]} be a square integrable martingale which is adapted to the filtration Ft defined above. Then, there exist U ∈ P² and Z ∈ P²(l²) such that

M_t=E[M_t] + _t

0

U_sdW_s+ ∞ i=1

_t

0

Z_s⁽ⁱ⁾dH_s⁽ⁱ⁾.

Proof. The Proof follows by combining the result of Løkka [5] (Theorem 5) and that of Nualart and Schoutens [6].

We denote by E the set ofR×R×l²–valued processes (Y, U, Z) defined onR+×Ω which areFt–adapted and such that:

(Y, U, Z)²=E

0≤t≤Tsup |Yt|²+ _T

0 |Us|²ds+ _T

0 Zs²ds

<+∞.

The couple (E,.) is then a Banach space.

We now introduce our BSDE. Given a data (f, ξ) we want to solve the following stochastic integral equation,which we denote by Equation (f, ξ):

Y_t=ξ+ _T

t

f(s, Y_s−, U_s, Z_s)ds− _T

t

U_sdW_s−^∞

i=1

_T

t

Z_s⁽ⁱ⁾dH_s⁽ⁱ⁾.

Definition 2.2: A solution of equationEq(f, ξ) is a triple (Y, U, Z) which belongs to the space (E,.) and satisfiesEq(f, ξ).

3 BSDE Driven by L ´ evy Processes

3.1 Existence and uniqueness of solutions

Theorem 3.1: Let the assumptions (A.1), (A.2) hold. Assume moreover that ξ is a square integrable random variable which isFT–measurable. Then Eq(f, ξ)has a unique solution.

Proof: Uniqueness. Let (Y, U, Z) and (Y , U , Z) be two solutions of equation Eq(f, ξ). By Itˆo’s formula 2.2,we have

E|Y_t−Y_t|²+E _T

t

|U_s−U_s|²ds+E _T

t

Z_s−Z_s²ds

= 2E _T

t

Y_s−−Y_s− f(s, Y_s−, U_s, Z_s)−f(s,Y_s−,U_s,Z_s)

ds, Sincef isL–Lipschitz,we get

E|Yt−Y_t|² +

1−2L β²

E

_T

t

|Us−U_s|²ds+

1−2L β²

E

_T

t

Zs−Z_s²ds

≤ L(β²+ 2)E _T

t

|Ys−−Y_s−|²ds,

where we have used the inequality 2xy≤β²x²+^y_β²₂. If we choose ^2L_β2 =¹₂,we obtain E|Yt−Y_t|²+E

_T

t

|Us−U_s|²ds+E _T

t

Zs−Z_s²ds≤CE _T

t

|Ys−Y_s|²ds.

Uniqueness now follows from Gronwall’s lemma.

Existence. Using the martingale representation theorem (Proposition 2.1),one can prove that the following BSDE

Y_t=ξ+ _T

t

f(s,0,0,0)ds− _T

t

U_sdW_s− _T

t

Zs, dH_s, has a solution.

Now,define (Yⁿ, Uⁿ, Zⁿ) as follows:

Y⁰=Z⁰=U⁰= 0 and (Yⁿ⁺¹, Uⁿ⁺¹, Zⁿ⁺¹) is the unique solution to the BSDE Y_tⁿ⁺¹=ξ+

_T

t

f(s, Y_s−ⁿ , U_sⁿ, Z_sⁿ)ds− _T

t

U_sⁿ⁺¹dW_s− _T

t

Z_sⁿ⁺¹, dH_s, We shall prove that (Yⁿ, Uⁿ, Zⁿ) is a Cauchy sequence in the Banach spaceE.

To simplify the notations,put :

Y^n,m_s :=Y_sⁿ−Y_s^m, U^n,m_s :=U_sⁿ−U_s^m and Z^n,m_s :=Z_sⁿ−Z_s^m and

f^n,m_s :=f(s, Y_s−ⁿ, U_sⁿ, Z_sⁿ)−f(s, Y_s−^m, U_s^m, Z_s^m).

Itˆo’s formula (2.2),shows that for everyn < m e^αt|Y^n+1,m+1_t |² +

_T

t

e^αs|U^n+1,m+1_s |²ds +

_T

t

e^αsZ^n+1,m+1_s ²ds+α _T

t

e^αs|Y^n+1,m+1_s− |²ds

= 2 _T

t

e^αsY^n+1,m+1_s− f^n,m_s ds−2 _T

t

e^αsY^n+1,m+1_s− U^n,m_s dW_s

−2 _T

t

e^αsY^n+1,m+1_s−

Z^n,m_s , dH_s

−(N_T −N_t),

where{N_t:t∈[0, T]} is a martingale given by N_t=

∞ i=1

∞ j=1

_t

0 e^αsZn+1,m+1,(i)

s Zn+1,m+1,(j)

s (d[H⁽ⁱ⁾, H^(j)]_s−d

H⁽ⁱ⁾, H^(j)

s).

Taking the expectation and using the fact thatH⁽ⁱ⁾, H^(j)=δ_i,jt,we get Ee^αt|Y^n+1,m+1_t |² +E

_T

t

e^αs|U^n+1,m+1_s |²ds +E

_T

t

e^αsZ^n+1,m+1_s ²ds+αE _T

t

e^αs|Y^n+1,m+1_s− |²ds

= 2E _T

t

e^αsY^n+1,m+1_s− f^n,m_s ds Sincef isL-Lipschitz,we get

e^αtE|Y^n+1,m+1_t |²+ _T

t

e^αsE|U^n+1,m+1_s |²ds

+ _T

t

e^αsEZ^n+1,m+1_s ²ds+α _T

t

e^αsE|Y^n+1,m+1_s− |²ds

≤2LE _T

t

e^αs|Y^n+1,m+1_s− |

|Y^n,m_s− |+|U^n,m_s |+Z^n,m_s ds, and then

e^αtE|Y^n+1,m+1_t |² + _T

t

e^αsE|U^n+1,m+1_s |²ds+ _T

t

e^αsEZ^n+1,m+1_s ²ds

+(α−L²β²) _T

t

e^αsE|Y^n+1,m+1_s− |²ds

≤ 3 β²E

_T

t

e^αs |Y^n,m_s− |²+|U^n,m_s |²+Z^n,m_s ² ds.

Choosingβ and αsuch that _β³2 = ¹₂ andα−6L²= 1, we get e^αtE|Y^n+1,m+1_t |² +

_T

t

e^αsE|U^n+1,m+1_s |²ds+ _T

t

e^αsEZ^n+1,m+1_s ²ds

≤1 2E

_T

t

e^αs |Y^n,m_s− |²+|U^n,m_s |²+Z^n,m_s ² ds It follows immediately,for allm > n,that

E _T

0

e^αs|Y^n,m_s− |²ds+E _T

0

e^αs|U^n,m_s |²ds+E _T

0

e^αsZ^n,m_s ²ds≤ C 2ⁿ.

Using again Itˆo’s formula and Doob’s inequality,it follows that there exists a universal constantC such that

E sup

0≤s≤T|Y^n,m_s |²+E _T

0 e^αs|U^n,m_s |²ds+E _T

0 e^αsZ^n,m_s ²ds≤ C 2ⁿ.

Consequently,(Yⁿ, Uⁿ, Zⁿ) is a Cauchy sequence in the Banach space E. It is not difficult to show that

(Y, U, Z) = lim

n→∞(Yⁿ, Uⁿ, Zⁿ), solves our BSDE.

The following theorem gives a bound for the difference between two solutions of Eq(f, ξ). It can be proved by using Itˆo’s formula,the Lipschitz property of f and Gronwall’s lemma.

Theorem 3.2: Given standard data(f, ξ)and(f ,ξ), let (Y, U, Z)and(Y , U , Z), be the unique solution the equationEq(f, ξ)andEq(f ,ξ) respectively. Then

E _T

0 |Y_s−−Y_s−|²+|U_s−U_s|²+Z_s−Z_s² ds

≤C

E|ξ−ξ|²+E _T

0 |f(s, Y_s−, U_s, Z_s)−f(s, Y_s−, U_s, Z_s)|²ds

.

3.2 Comparison theorem

In this subsection,we prove a comparison theorem for BSDE driven by L´evy process.

This is an important tool in the probabilistic interpretation of viscosity solutions of partial differential equations.

Theorem 3.3: Given standard data(f₁, ξ₁) and(f₂, ξ₂), suppose that ξ₁≤ξ₂ and f₁(t, y, u, z)≤f₂(t, y, u, z) for all(y, u, z)∈R×R×l²,dP×dt–a.s. ThenY_t^f1 ≤Y_t^f2, t∈[0, T].

Proof: Set

Y_s:=Y_s^f2−Y_s^f1, U_s:=U_s^f2−U_s^f1, Z_s:=Z_s^f2−Z_s^f1, ξ:=ξ₂−ξ₁,

and

f_s:=f₂(s, Y_s−^f2, U_s^f2, Z_s^f2)−f₁(s, Y_s−^f2, U_s^f2, Z_s^f2).

We define three stochastic processes as follows α_s=

Y⁻¹_s− f₁(s, Y_s−^f2, U_s^f2, Z_s^f2)−f₁(s, Y_s−^f1, U_s^f2, Z_s^f2)

ifY_s−= 0

0 ifY_s−= 0,

β_s=

U⁻¹_s f₁(s, Y_s−^f1, U_s^f2, Z_s^f2)−f₁(s, Y_s−^f1, U_s^f1, Z_s^f2)

ifU_s= 0

0 ifU_s= 0,

and for all i∈N^∗ letZ⁽ⁱ⁾ denote thel²–valued stochastic process such that itsi first components are equal to those ofZ^f2 and itsN^∗\{1,2, . . . , i}last components are equal to those ofZ^f1. With this notation,we define fori∈N^∗

γ_s⁽ⁱ⁾=





(Z⁽ⁱ⁾_s )⁻¹ f₁(s, Y_s−^f1, U_s^f1,Z_s⁽ⁱ⁾)−f₁(s, Y_s−^f1, U_s^f1,Z_s⁽ⁱ⁻¹⁾)

ifZ⁽ⁱ⁾_s = 0

0 ifZ⁽ⁱ⁾_s = 0.

It is clear that

γ_s, Z_s

= f₁(s, Y_s−^f1, U_s^f1, Z_s^f2)−f₁(s, Y_s−^f1, U_s^f1, Z_s^f1)

,

and the processes{αt:t∈[0, T]},{βt:t∈[0, T]}and{γt:t∈[0, T]}are progressively measurable and bounded.

For 0≤s≤t≤T,let

M_t^H,W :=

_t

0 β_sdW_s+ _t

0 γ_s, dH_s Γ_s,t:= exp

_t

s

α_rdr−d[M_.^H,W]^c_r+dM_r^H,W

s<r≤t

1 + ∆M_r^H,W exp

−∆M_r^H,W . Using Itˆo’s formula (2.1) one can see that{Γ_s,r:r∈[s, T]}satisfies the stochastic linear equation

Γ_s,t= 1 + _t

s

Γ_s,r−dM_r^H,W + _t

s

Γ_s,r−α_rdr. (3.1) Since

Y_t = ξ+ _T

t

α_rY_r−+β_rU_r+

γ_r, Z_r dr +

_T

t

f_rdr− _T

t

U_rdW_s− _T

t

Z_r, dH_r , we use formula (2.3) and relation (3.1) to show that for all 0≤s≤t≤T

Y_s = Γ_s,tY_t+ _t

s

Γ_s,r−f_rdr

− _t

s

Γ_s,r−

U_r+β_rY_r−

dW_s− _t

s

Γ_s,r−

γ_rY_r−+Z_r, dH_r +

∞ i=1

_t

s

Γ_s,r−γ_r⁽ⁱ⁾Z⁽ⁱ⁾_r (d[H⁽ⁱ⁾]_r−d < H⁽ⁱ⁾>_r).

Since the last three terms in the right–hand of the above equation are martingales,we deduce that

Y_s=E

Γ_s,tY_t+ _t

s

Γ_s,r−f_rdr /Fs

. Hence,the result follows,fort=T,by the positivity ofξ andf.

4 BSDE with Locally Lipschitz Coefficient

The aim of this section is to prove the existence and uniqueness of solutions for BSDE with locally Lipschitz generator. More precisely,we assume that the following conditions hold:

H.1) f is continuous in (y, u, z) for almost all (t, ω),

H.2) there existsK >0 and 0≤α <1 such that|f(t, ω, u, y, z)| ≤K(1 +|y|^α+|u|^α+ z^α).

H.3) for everyN∈N,there exists a constant L_N >0 such that

|f(t, ω, y, u, z)−f(t, ω, y, u, z)| ≤L_N(|y−y|+|u−u|+z−z), P–a.s.,a.e.

t∈[0, T]

and∀y,y,u,u,z,z such that |y| ≤N,|y| ≤N,|u| ≤N,|u| ≤N,z ≤N, z ≤N.

When the assumptions H.1) and H.2) are satisfied,we can define the family of semi–

norms (ρ_n(f))_n

ρ_n(f) =

E _T

0 sup

|y|,|u|, z ≤n|f(s, y, u, z)|²ds ¹₂

.

We denote byLip_loc (resp. Lip) the set of processesf satisfying H.1)–H.2) which are locally Lipschitz,i.e. satisfy the assumption H.3),(resp. globally Lipschitz) with respect to (y, u, z).

Lip_loc,αdenotes the subset of those processesf which belong toLip_locand which satisfy H.2).

The main results are the following

Theorem 4.1: (Existence and uniqueness). Let f ∈ Lip_loc,α and ξ be a square integrable random variable. Then equation Eq(f, ξ) has a unique solution if L_N ≤ L+

log(N), whereL is some positive constant.

We give now a stability result for the solution with respect to the data (f, ξ). Roughly speaking,iff_n converges tof in the metric defined by the family of semi–norms (ρ_N) andξ_nconverges toξinL²(Ω) then (Yⁿ, Uⁿ, Zⁿ) converges to (Y, U, Z) inE. Let (f_n) be a sequence of functions which areF_t–progressively measurable for eachn. Let (ξ_n)_n≥1 be a sequence of random variables which are FT–measurable for eachnand such that E|ξn|² < ∞. We will assume that for eachn,the BSDE Eq(f_n, ξ_n) corresponding to the data (f_n, ξ_n) has a (not necessarily unique) solution. Each solution of the equation Eq(f_n, ξ_n) will be denoted by (Y^fn, Z^fn).

We suppose also that the following assumptions H.4),H.5) and H.6) are fulfilled, H.4) For every N,ρ_N(f_n−f)−→0 asn→ ∞.

H.5) E|ξn−ξ|²−→0 asn→ ∞.

H.6) There existK >0 such that, sup

n |f_n(t, ω, y, u, z)| ≤K(1 +|y|^α+|u|^α+z^α) P–a.s., a.e. t∈[0, T].

Theorem 4.2:(Stability). Let f andξ be as in Theorem4.1. Assume that (f_n, ξ_n) satisfiesH.4),H.5) andH.6). Then we have

n→+∞lim

E sup

0≤t≤T|Y_t^fn−Y_t|²+E _T

0 |U_s^fn−U_s|²ds+E _T

0 Z_s^fn−Z_s²ds

= 0.

To prove Theorems 4.1 and 4.2 we need the two following lemmas.

Lemma 4.3: Let ξ¹, ξ² be two d–dimensional square integrable random variables which are FT–measurable. Let f₁ and f₂ be two functions which satisfy H.1), H.2).

Let (Y^f1, U^f1, Z^f1) [resp. (Y^f2, U^f2, Z^f2)] be a solution of the BSDE Eq(f₁, ξ¹) [resp.

Eq(f₂, ξ²)]. Then for every locally Lipschitz function f and everyN >1, the following estimates hold

E _T

0 |U_s^f1−U_s^f2|²ds+E _T

0 Z_s^f1−Z_s^f22ds

≤ C(K, ξ¹, ξ²)



E(|ξ¹−ξ²|²) +

E _T

0 |Y_s^f1−Y_s^f2|²ds ¹₂





and

E(|Y_s^f1−Y_s^f2|²) ≤ C₁!

E(|ξ¹−ξ²|²) +ρ²_N(f₁−f) +ρ²_N(f−f₂)

+ C(K, ξ¹, ξ²) (2L_N + 2L²_N)N^2(1−α)

"

exp

(2L_N+ 2L²_N)(T−s) , where C(K, ξ¹, ξ²) is a constant which depends onK, E|ξ¹|² and E|ξ²|², and C₁ is a universal constant.

Proof: The first inequality follows from Itˆo’s formula and Schwarz inequality. We shall prove the second one. Let<, >denote the inner product inR^d.

We set

Y_s:=Y_s^f1−Y_s^f2, U_s:=U_s^f1−U_s^f2 and Z_s:=Z_s^f1−Z_s^f2, and

f_s:=f₁(s, Y_s−^f1, U_s^f1, Z_s^f1)−f₂(s, Y_s−^f2, U_s^f2, Z_s^f2).

By Itˆo’s formula we have

##Y_t##² + _T

t

##U_s##²ds+ _T

t

Zs²ds=|ξ¹−ξ²|²+ 2 _T

t

Y_s−f_sds−2 _T

t

Y_s−U_sdW_s

−2^∞

i=1

_T

t

Y_s−Z⁽ⁱ⁾_s dH_s⁽ⁱ⁾−^∞

i=1

∞ j=1

_T

t

Z⁽ⁱ⁾_s Z^(j)_s d [H⁽ⁱ⁾, H^(j)]_s−< H⁽ⁱ⁾, H^(j)>_s

.

Using the fact that _t

0Z⁽ⁱ⁾_s Z^(j)_s d

[H⁽ⁱ⁾, H^(j)]_s−< H⁽ⁱ⁾, H^(j)>_s

is a martingale and taking the expectation we get

E##Y_t##²+E _T

t

##U_s##²ds+E _T

t

Zsds=E|ξ¹−ξ²|²+ 2E _T

t

Y_s−, f_sds.

Letβ andγ be strictly positive numbers. For a givenN >1,letL_N be the Lipschitz constant off in the ballB(0, N),

A^N :=

$

(s, ω); |Y_s−^f1|²+|U_s^f2|²+Z_s^f12+|U_s^f1|²+|Y_s−^f2|²+Z_s^f22≥N²

% , A^N,c:= Ω\A^N and denote by 11_A the indicator function of the setA. We have E|Yt|²+E

_T

t

|Us|²ds+E _T

t

Zs²ds = E|ξ¹−ξ²|²+ 2E _T

t

Ys−, f_s(11_A^N+ 11_AN,c)ds := E|ξ¹−ξ²|²+I₁+I₂+I₃+I₄,

where

I₁:= 2E _T

t

Y_s−, f_s11_ANds I₂:= 2E

_T

t

Y_s−, f₁(s, Y_s−^f1, U_s^f1, Z_s^f1)−f(s, Y_s−^f1, U_s^f1, Z_s^f1)11_A^N,cds I₃:= 2E

_T

t

Y_s−, f(s, Y_s−^f1, U_s^f1, Z_s^f1)−f(s, Y_s−^f2, U_s^f2, Z_s^f2)11_A^N,cds.

I₄:= 2E _T

t

Ys−, f(s, Y_s−^f2, U_s^f2, Z_s^f2)−f₂(s, Y_s−^f2, U_s^f2, Z_s^f2)11_A^N,cds.

It is not difficult to check that I₂ ≤ E

_T

t

|Y_s−|²11_AN,cds+ρ²_N(f₁−f) I₄ ≤ E

_T

t

|Y_s−|²11_AN,cds+ρ²_N(f−f₂).

Sincef isL_N-Lipschitz in the ballB(0, N),we get I₃≤(2L_N +γ²)E

_T

t

|Y_s−|²11_AN,cds+2L²_N γ² E

_T

t

|U_s|²ds+2L²_N γ² E

_T

t

Z_s²ds.

To estimateI₁,we use H¨older’s inequality and the fact that

11_AN ≤ |Y_s−^f1|²+|U_s^f2|²+Z_s^f12+|U_s^f1|²+|Y_s−^f2|²+Z_s^f22 N²

to obtain I₁ ≤ β²E

_T

t

|Ys|²11_ANds+ 1 β²E

_T

t

|f_s|²11_ANds