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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 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.

Keywords

Reflected backward stochastic differential equationpenalization methodoptimal stoppingSnell envelopeDynkin game.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 3 - 22
Copyright
© EDP Sciences, SMAI, 2007

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References

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