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Reflected backward stochastic differential equations with two RCLL barriers

Published online by Cambridge University Press:  01 March 2007

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

In this paper we consider BSDEs with Lipschitzcoefficient reflected on two discontinuous (RCLL) barriers. In thiscase, we prove first the existence and uniqueness of the solution,then we also prove the convergence of the solutions of the penalizedequations to the solution of the RBSDE. Since the method used in thecase 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 notwork, we develop a new method, by considering the solutions of thepenalized equations as the solutions of special RBSDEs and usingsome results of Peng and Xu in Annales of I.H.P.41 (2005) 605–630.

Keywords

Reflected backward stochasticdifferential equationpenalization methodoptimal stoppingSnellenvelopeDynkin game.

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References

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