Reflected BSDEs driven by inhomogeneous simple Lévy processes with RCLL barrier

Abstract

In this paper, we study the solution of a backward stochastic differential equation driven by an inhomogeneous simple Lévy process with a rcll reflecting barrier. We show the existence and uniqueness of solution by means of the Snell envelope and the fixed point theorem when the coefficient is stochastic Lipschitz. In term of application, we provide the fair price of the American option in Lévy market. Introduction. The theory of backward stochastic differential equations (BSDEs for short) was developed by Pardoux and Peng [25]. These equations have attracted great interest due to their connections with mathematical finance [10, 8], stochastic control and stochastic games [7, 15, 17, 16]. There have been many studies done on this topic lately, and we can’t talk about those studies without mentioning the Situ’s one [29], which was on BSDEs driven by a Brownian motion and a Poisson point process. In addition to the study of Nualart and Schoutens [24] who have established the existence and uniqueness of solutions for BSDEs driven by a Lévy process. Also, the great study of Bahlali et al. [1] in which they treated the case where the BSDE is driven by a Brownian motion and the martingales of Teugels associated with an independent Lévy process. And last but not least, El Jamali and El Otmani’s [5] in which we have established the existence and uniqueness of solutions for BSDEs driven by an inhomogeneous Lévy processes when the coefficient is stochastic Lipschitz. In the framework of a Brownian filtration, the notion of reflected BSDE has been introduced by ElKaroui et al. [11]. A solution of such an equation that is associated with a coefficient f , terminal value ξ and a barrier L, is a triple process (Y,Z,K) Achieving:  Yt = ξ + ∫ T t f(s, Ys, Zs)ds+KT −Kt − ∫ T t ZsdBs. Yt ≥ Lt P− a.s. for all t ≤ T. The role of the continuous increasing process K is to push upwards the process Y in order to keep it above the barrier with minimal energy, that is, ∫ T 0 (Yt − Lt)dKt = 0. This type of BSDEs is motivated by pricing the American options [9] and studying the mixed game problems [18]. The extension to the cases of reflected BSDE with jumps, which are first, a standard reflected BSDE driven by a Brownian motion and an independent Poisson point process, has been established by Hamadène and Ouknine [19]. Second, Essaky’s [13] studied on the reflected BSDEs with jumps and right continuous left hand limited (rcll for short) obstacle. Third, El Otmani [12] has considered a reflected BSDE driven by a Brownian motion and the martingales of Teugels associated with a pure jump independent Lévy process and rcll obstacle (see e.g. [14, 27, 30]). And last but not least, Lü [23] who treated the case where the reflected BSDE driven by a Brownian motion and the martingales 1991 AMS Mathematics subject classification. 60H20, 60H30, 60J75, 65C30.