Reflected BSDEs Driven by Lévy Processes and Countable Brownian Motions

A new class of reflected backward stochastic differential equations (RBSDEs) driven by Teugels martingales associated with Lévy process and Countable Brownian Motions are investigated. Via approximation, the existence and uniqueness of solution to this kind of RBSDEs are obtained.


Introduction
Recently, Y. Ren [1] proved via the Snell envelope and the fixed point theorem, the existence and uniqueness of a solution for the following RBDSDEs driven by a Lévy process and a extra Brownian motion with Lipschitz coefficients, where the obstacle process is right continuous with left limits (càdlàg): where the d i H is a forward semi-martingale Itô integrals (see He et al. [2]) and the dB  is a backward Itô integral.
Note that, in all the previous works, the equations are driven by finite Brownian motions.In their recent work, Pengju Duan et al. [3] introduced firstly the reflected BDSDEs driven by countable extra Brownian motions: where the dW is the standard forward stochastic Itô integral and the d j B  is the backward stochastic Itô integral.
Under the global Lipschitz continuity conditions on the coefficients f and g, they proved via Snell envelope and fixed point theorem, the existence and uniqueness of the solution for RBDSDEs (1.1).Next, J.-M.Owo [4] relaxed the Lipschitz continuity condition on the coefficient f to a continuity with sub linear growth condition and derive the existence of minimal and maximal solutions to RBSDEs (1.1).
Motivated by [1] [3] [4], in this paper, we mainly consider the following RBDSDEs driven by a Lévy process and countable Brownian motions, in which the obstacle process is right continuous with left limits (càdlàg): The paper is devoted to prove the existence and uniqueness of a solution for RBSDEs driven by a Lévy process and countable Brownian motions.The paper is organized as follows.In section 2, we give some preliminaries and notations.In section 3, we establish the main results.

Preliminaries and Notations
Throughout this paper, T is a positive constant and ( ) where for any process { } is neither increasing nor decreasing so it does not constitute a filtration.Let us introduce some spaces:  denotes the space of real-valued processes { } [ ]  formed by the  -predictable processes; • 2   stands for the set of real-valued, càdàg, random processes { }  denotes the space continuous, real-valued, increasing processes { }  denotes the set of real valued sequences ( ) We will denote by ( ) ( ) In the sequel, for ease of notation, we set , L are power-jump processes.That is, ( ) In [5], Nualart and Schoutens proved that the coefficients , i k c correspond to the orthonormalization of the polynomials 2 1, , , x x ⋅⋅⋅ with respect to the measure ( ) ( ) ( ) ( ) can be chosen to be pairwise strongly orthonormal martingale.That is, for all , i j , Definition 2.1.A solution of a (1.2) is a triplet of ( ) , , Y Z K , which satisfies (1.2), and 1) ( ) ( ) , , ; K is a continuous and increasing process with 0 0 K = and ( ) Throughout the paper, we let the coefficients →  satisfying the following assumptions: , , , , , , [ ] , where . Moreover, we assume that its jumping times are inaccessible stopping times (see He et al. [2]).

The Main Results
We first establish the existence and uniqueness result for RBSDEs driven by finite Brownian motions and a Lévy process: For any 1 n ≥ , we have the following existence and uniqueness result.

Lemma 3.2. Assume (H1) -(H4).
Then, there exists a unique solution ( ) , , Y Z K of Equation (3.1).Proof.For 1 n = , we obtain the existence and uniqueness result due to Y. Ren [1].For any 1 n > , we can prove the desired result following the same ideas and arguments as in Y. Ren [1]: it is a straightforward adaptation of the proofs of Theorem 2 and Theorem 3 in Y. Ren [1].Firstly, we consider the special case that is the function f and j g do not depend on (Y, Z), i.e.
, , , , . Therefore, we omit the details.Now, we are ready to establish the main result of this paper which is the following theorem.
Theorem 3.3.Under assumptions (H1)-(H4), there exists a unique solution ( ) ( ) Proof.(Existence.)By Lemma 3.1, for any 1 n ≥ , there exists a unique solution of (3.1), denoted by ( ) The idea consists to study the convergence of the sequence ( ) Y Z K , and to establish that its limit is a solution of (1.2).To this end, we first establish the following estimates: where λ is a non-negative constant independent of n.Indeed, applying Itô's formula to From assumption (H2) and Young's inequality, for any 0, , , , , ,0,0 , 2 ,0,0 .
Using again Young inequality, we have for any 0 β > , ( ) ,0,0 d We choose , 0 Therefore, we have the existence of a constant 1 c such that ( ) which by Burkhölder-Davis-Gundy's inequality provides    .To this end, without loss of generality, we let m n < .Then, by difference, we obtain .
Consequently, ( )    which is a Banach space.Therefore, there exists a process ( ) ( ) Now, let us show that the process ( ) ( ) Also, by Burkhölder-Davis-Gundy's inequality, we get Now, from (H1)-(H2) and the fact that ( ) ( ) On the other hand, from the result of Saisho [6]    and assumption (H2), we obtain, ( ) ] we have .Then, we complete the proof.
Let  denote the class of P-null sets of  .For each[ ] associated with the Lévy process { } suffices to replace suitably 0, , , ,