Reflected backward stochastic differential equations with two RCLL barriers

Issue		ESAIM: PS Volume 11, 2007


Page(s)		3 - 22
DOI		10.1051/ps:2007002
Published online		01 March 2007

ESAIM: P&S, February 2007, Vol. 11, pp. 3-22
DOI: 10.1051/ps:2007002

Reflected backward stochastic differential equations with two RCLL barriers

Jean-Pierre Lepeltier and Mingyu Xu

Département de Mathématiques, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France; Jean-Pierre.Lepeltier@univ-lemans.fr; xvmingyu@hotmail.com

(Invited paper accepted September 2005. Published online 1 March 2007.)

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.

Mathematics Subject Classification. 60H10, 60G40.

Key words: Reflected backward stochastic differential equation, penalization method, optimal stopping, Snell envelope, Dynkin game.

© EDP Sciences, SMAI 2007

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