Multidimensional BSDE with Poisson jumps of Osgood type


 This paper is devoted to solve a multidimensional backward stochastic differential equation with jumps in finite time horizon. Under linear growth generator, we prove existence and uniqueness of solution.


Introduction
It is well known that Backward stochastic differential equations (BSDEs in short) driven by random Poisson measure are natural extension of classical BSDEs. These equations, first discussed by Tang and Li [8] can be seen as a generalization of Pardoux and Peng's work [6], which constitute the key point of solving problem in financial mathematics and studying non linear partial differential equations (PDEs in short) by means of stochastic tools. Since then the interest in searching probabilistic formula of solution of other type of PDEs increases a lot. Some authors studying parabolic integral-partial differential equation (PIDE), interested in BSDEs with Poisson Process (BSDEP in short). Among them we mention the result of Barles et al [1] who establish a probabilistic interpretation of a solution of a PIDE. By means of a comparison theorem, they generalized the probabilistic representation of solution of quasilinear PDEs proved in [6] to PIDEs. But all these results are obtained either with a Lipschitz condition or a monotonicity one on the drift of the stochastic equation. Several authors investigate in weakening this restrictive assumption. Among others Mao [3] investigate successfully these equations with the Osgood condition. This one is introduced by specific function which allows the use of the well known Bihari's Lemma to get uniqueness. The limitation is that all these results are established in the one dimensional case.
The study of multidimensional BSDEs with weak conditions on the generator was discuss recently by Fan et al [2]. Using a suitable sequence, they prove an existence and uniqueness result when the generator satisfies the Osgood condition. In this work we interested in extending this result to multidimensional BSDEs driven by random Poisson measure (MBSDEPs in short) satisfying the Osgood condition. Inspired by the method introduced by Fan et al [2], we prove existence and uniqueness of solution of a MBSDEP. The paper is organized as follows. In section 2, we recall some important results on MBSDEs driven by Poisson random measure. In section 3, we establish our main result.

Definitions and preliminary results
Let Ω be a non-empty set, F a σ −algebra of sets of Ω and P a probability measure defined on F . The triplet (Ω, F , P) defines a probability space, which is assumed to be complete. We assume given two mutually independent processes : We consider the filtration (F t ) t≥0 given by Here N denotes the class of P−null sets of F . For Q ∈ N * , | . | stands for the Euclidian norm in R Q . We consider the following sets (where E denotes the mathematical expectation with respect to the probability measure P), and a non-random horizon time 0 < T < +∞: We may often write | · | instead of · L 2 (E,E ,λ ) for a sake of simplicity.
Let k ≥ 1 and define is a Banach space. Finally let S be the set of all non-decreasing and concave function ϕ(·) : where Θ s stands for the triple (Y s , Z s ,U s ).
For instance let us precise the notion of solution to eq.(2.1).
Now, let us introduce the following Proposition 2.2, which will play an important role in the proof of Theorem 3.4. We consider the following assumption on the generator f : y, f (ω,t, y, z, u) ≤ γ(t)ψ(|y| 2 ) + α(t)|y| (|z| + |u|) + |y|ϕ t where γ, α : [0, T ] → R + satisfying 0 <´T 0 [α 2 (s) + γ(s)]ds < ∞, ϕ ∈ M 2 (R) is non-negative and ψ is nondecreasing concave function from R + to itself with ψ(0) = 0. Proposition 2.2. Assume that f satisfies (A) and let (Y t , Z t ,U t ) 0≤t≤T be a solution to the MBSDEP (2.1). There exists a constant c > 0 depending only on α such that, for any 0 ≤ t ≤ T , Proof. Itô's formula applied to |Y t | 2 reads to By the assumption (A) and the inequality 2ab ≤ θ a 2 + b 2 /θ for any θ > 0, we have where for 0 ≤ t ≤ T , Applying Burkhölder-Davis-Gundy inequality, we derive that the process M t =´t 0 Y s , Z s dB s 0≤t≤T is in fact a uniformly integrable martingale and there exists δ > 0 such that for 0 ≤ t ≤ T , we have Similarly for the discontinuous martingale, there exists δ > 0 such that for 0 ≤ t ≤ T , we have (2.7) By virtue of (2.5), (2.6) and (2.7), we deduce from eq.
Hence combining the above inequality with (2.5), we deduce that there exists a constant c > 0 depending only on δ such that where g(t) stands for the left hand side of (2.8).
We are in position to investigate our main result.

Existence and uniqueness of solution
Let us introduce the following assumptions on the generator f . We say that f satisfies assumptions (H1) if the following hold (were we define for 0 ≤ s ≤ T, f (s, 0) = f (s, 0, 0, 0) to ease the reading): • (H1.1): f satisfies the weak Lipschitz condition in y, i.e., there exists ρ ∈ S such that dP × dt-a.e, ∀(y, y ) ∈ • (H1. 2): f is Lipschitz continuous in (z, u) uniformly with respect to (ω,t, y), i.e., there exists a function β : The integrability condition holds a.s.
We recall the following results, which will be useful in the proof of uniqueness. As in [2], Theorem 1, we consider the sequence ( f n ) n≥1 defined by We have the following result whose proof is omitted since it is an adaptation of step 1 of Theorem 1 in [2].
T h i s p a g e i s i n t e n t i o n a l l y l e f t b l a n k