Reflected backward stochastic differential equations with two RCLL barriers

Abstract

In this paper we consider BSDEs with Lipschitz coefficient reflected on two discontinuous (RCLL) barriers. In this case, we prove first the existence and uniqueness of the solution, then we also prove the convergence of the solutions of the penalized equations to the solution of the RBSDE. Since the method used in the case of continuous barriers (see Cvitanic and Karatzas, Ann. Probab. 24 (1996) 2024-2056 and Lepeltier and San Martín, J. Appl. Probab. 41 (2004) 162-175) does not work, we develop a new method, by considering the solutions of the penalized equations as the solutions of special RBSDEs and using some results of Peng and Xu in Annales of I.H.P. 41 (2005) 605-630. © EDP Sciences, SMAI 2007.