BSDE ASSOCIATED WITH LÉVY PROCESSES AND APPLICATION TO PDIE

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 √ log(N) in the ball B(0, N), then the corresponding BSDE 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).


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 (Hamadène 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: (1.1) When the coefficient f is uniformly Lipschitz, the BSDE (1.1) has a unique solution.The proof is mainly based on the Itô martingale representation theorem.
In Nualart and Schoutens [6], a martingale representation theorem associated to Lévy processes was proved.It then is natural to extend equations (1.1) to BSDE's driven by a Lévy 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 Lévy 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 ball B(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 Lévy 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 Lévy 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.
We assume that where N denotes the totality of P-null sets and Let H 2 denote the space of real valued, square integrable and F t -progressively measurable processes φ = {φ t : t ∈ [0, T ]} such that and denote by P 2 the subspace of H 2 formed by the predictable processes.Let l 2 be the space of real valued sequences (x n ) n≥0 such that ∞ i=0 x 2 i is finite.We shall denote by H 2 (l 2 ) and P 2 (l 2 ) the corresponding spaces of l 2 -valued processes equipped with the norm

Let us define: (
We recall the Itô formula for càdlàg semimartingales.

Itô's formula
Let X = {X t : t ∈ [0, T ]} be a càdlàg semimartingale, with quadratic variation denoted by [X] = {[X] t : t ∈ [0, T ]} and let F be a C 2 real valued function.Then F (X) is also a semimartingale and the following formula holds: where [X] c (sometimes denoted by X ) is the continuous part of the quadratic variation [X].We also note that in the case where F (x) = x 2 , the formula (2.1) takes the form Moreover if X and Y are two càdlàg semimartingales then we have where [X, Y ] stands for the quadratic covariation of X , Y also called the bracket process.For a complete survey in this topic we refer to Protter [11].

Predictable representation
We denote by (H (i) ) i≥1 the Teugel's Martingales associated with the Lévy process {L t : t ∈ [0, T ]}.More precisely where 1 ] for all i ≥ 1 and L (i) t are power-jump processes.That is, L It was shown in Nualart and Schoutens [6] that the coefficients c i,k correspond to the orthonormalization of the polynomials 1, x, x 2 , ... with respect to the measure µ(dx) = x 2 ν(dx) + σ 2 δ 0 (dx): We set The martingales (H (i) ) i≥1 can be chosen to be pairwise strongly orthonormal martingales.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 F t defined above.Then, there exist U ∈ P 2 and Z ∈ P 2 (l 2 ) such that 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 of R × R × l 2 -valued processes (Y, U, Z) defined on R + × Ω which are F t -adapted and such that: 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):

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

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 is F T -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 Itô's formula 2.2, we have where we have used the inequality 2xy Uniqueness now follows from Gronwall's lemma.
Existence.Using the martingale representation theorem (Proposition 2.1), one can prove that the following BSDE To simplify the notations, put : Itô's formula (2.2), shows that for every n < m where {N t : t ∈ [0, T ]} is a martingale given by Taking the expectation and using the fact that Using again Itô's formula and Doob's inequality, it follows that there exists a universal constant C such that 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 Itô's formula, the Lipschitz property of f and Gronwall's lemma.

Comparison theorem
In this subsection, we prove a comparison theorem for BSDE driven by Lévy process.This is an important tool in the probabilistic interpretation of viscosity solutions of partial differential equations.Theorem 3.3: Proof: Set and ).We define three stochastic processes as follows and for all i ∈ N * let Z (i) denote the l 2 -valued stochastic process such that its i first components are equal to those of Z f 2 and its N * \{1, 2, . . ., i} last components are equal to those of Z f 1 .With this notation, we define for i ∈ N * Since we use formula (2.3) and relation (3.1) to show that for all 0 Since the last three terms in the right-hand of the above equation are martingales, we deduce that Hence, the result follows, for t = T , by the positivity of ξ and f .

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.3) for every N ∈ N, there exists a constant When the assumptions H.1) and H. .
We denote by Lip loc (resp.Lip) the set of processes f 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,α denotes the subset of those processes f which belong to Lip loc and 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 ), where L is some positive constant.
We give now a stability result for the solution with respect to the data (f, ξ).Roughly speaking, if f n converges to f in the metric defined by the family of semi-norms (ρ N ) and ξ n converges to ξ in L 2 (Ω) then (Y n , U n , Z n ) converges to (Y, U, Z) in E. Let (f n ) be a sequence of functions which are F t -progressively measurable for each n.Let (ξ n ) n≥1 be a sequence of random variables which are F T -measurable for each n and such that E|ξ n | 2 < ∞.We will assume that for each n, 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 f n , Z f n ).
We suppose also that the following assumptions H.4), H.5) and H.6) are fulfilled, To Then for every locally Lipschitz function f and every N > 1, the following estimates hold where C(K, ξ 1 , ξ 2 ) is a constant which depends on K, E|ξ 1 | 2 and E|ξ 2 | 2 , and C 1 is a universal constant.Proof: The first inequality follows from Itô's formula and Schwarz inequality.We shall prove the second one.Let <, > denote the inner product in R d .We set and . By Itô's formula we have Using the fact that ) , H (j) ] s − < H (i) , H (j) > s is a martingale and taking the expectation we get Let β and γ be strictly positive numbers.For a given N > 1, let L N be the Lipschitz constant of f in the ball B(0, N), A N,c := Ω \ A N and denote by 1 1 A the indicator function of the set A. We have where It is not difficult to check that Since f is L N -Lipschitz in the ball B(0, N), we get To estimate I 1 , we use Hölder's inequality and the fact that to obtain If we choose β 2 = 2L 2 N + 2L N and γ 2 = 2L 2 N then we use the above estimates we have Using Gronwall's lemma, we get Proof: Let ψ n be a sequence of smooth functions with support in the ball B(0, n+1) and such that ψ n = 1 in the ball B(0, n).It is not difficult to see that the sequence (f n ) of truncated functions, defined by f n = fψ n , satisfies all the properties quoted in Lemma 4.4.
(ii) There exists a process (Y, U, Z) ∈ E such that Proof of Lemma 4.5: For simplicity,we assume L = 0. Assertion (i) follows from standard arguments of BSDE.Let us prove (ii).First, assume that L N ≤ f m , ξ) and next passing to the limits successively on n, m, N one gets Lemma 4.5.Assume now that L N ≤ log(N ).Let δ be a strictly positive number such that δ < (1−α)  2 .Let ([t i+1 , t i ]) be a subdivision of [0, T ] such that |t i+1 − t i | ≤ δ.Applying Lemma 4.3 in all the subintervals [t i+1 , t i ] we get Lemma 4.5.
Proof of Theorems 4.1 and 4.2:.The uniqueness follows from Lemma 4.3 by letting f 1 = f 2 = f and ξ 1 = ξ 2 = ξ).We shall prove the existence of solutions.By Lemma 4.5, there exists (Y, U, Z) It remains to prove that Since f is L N -locally Lipschitz, we use the triangle inequality and Lemma 4.4 to obtain Since |x| α ≤ 1 + |x| for each α ∈ [0, 1[, we successively use Lemma 4.5 (i) -b), Schwarz inequality, Chebychev inequality, Lemma 4.5 (i) and Fatou's lemma to get where K 1 , K 2 denote the two constant defined in Lemma 4.5 (ii) and where and by passing to the limits, first on n and next on N .The proofs are finished.

Applications to PDIE
In this section, we give the links between BSDE driven by Lévy process and a family of partial differential integral equation (PDIE).Let X t = t 0 σ(X s )dW s + L t , recall that L t is a Lévy process with Lévy measure ν, which takes the form L t = bt + t .
We give a technical lemma which will be needed later on.dH (1)  s , from which we get the desired result.

Lemma 4 . 5 :
Let f and ξ be as in Theorem 4.1.Let (f n ) be the sequence of functions associated to f by Lemma 4.4 and denote by
prove Theorems 4.1 and 4.2 we need the two following lemmas.Lemma 4.3: Let ξ 1 , ξ 2 be two d-dimensional square integrable random variables which are F T -measurable.Let f 1 and f 2 be two functions which satisfy H.1), H.2).