Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-11T05:57:39.870Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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.
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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M. Alario-Nazaret, Jeux de Dynkin. Ph.D. dissertation, Univ. Franche-Comté, Besançon (1982).
Alario-Nazaret, M., Lepeltier, J.P. and Marchal, B., Dynkin games. Lect. Notes Control Inform. Sci. 43 (1982) 23–42. CrossRef
Bismut, J.M., Sur un problème de Dynkin. Z.Wahrsch. Verw. Gebiete 39 (1977) 31–53. CrossRef
Cvitanic, J. and Karatzas, I., Backward Stochastic Differential Equations with Reflection and Dynkin Games. Ann. Probab. 24 (1996) 2024–2056.
El Karoui, N., Les aspects probabilistes du contrôle stochastique, in P.L. Hennequin Ed., Ecole d'été de Saint-Flour. Lect. Notes Math. 876 (1979) 73–238. CrossRef
El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M.C., Reflected Solutions of Backward SDE and Related Obstacle Problems for PDEs. Ann. Probab. 25 (1997) 702–737.
Hamadène, S., Reflected BSDE's with Discontinuous Barrier and Application. Stochastics and Stochastic Reports 74 (2002) 571–596. CrossRef
Lepeltier, J.P. and San Martín, J., Backward SDE's with two barriers and continuous coefficient. An existence result. J. Appl. Probab. 41 (2004) 162–175. CrossRef
Lepeltier, J.P. and Penalization, M. Xu method for Reflected Backward Stochastic Differential Equations with one RCLL barrier. Statistics Probab. Lett. 75 (2005) 58–66. CrossRef
Pardoux, E. and Peng, S., Adapted solutions of Backward Stochastic Differential Equations. Systems Control Lett. 14 (1990) 51–61. CrossRef
Peng, S. and Smallest, M. Xu g-Supermartingales and related Reflected BSDEs. Annales of I.H.P. 41 (2005) 605–630.