Reflected backward stochastic differential equations with two RCLL barriers
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 3-22.

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.

DOI : 10.1051/ps:2007002
Classification : 60H10, 60G40
Mots-clés : reflected backward stochastic differential equation, penalization method, optimal stopping, Snell envelope, Dynkin game 
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     title = {Reflected backward stochastic differential equations with two {RCLL} barriers},
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     pages = {3--22},
     publisher = {EDP-Sciences},
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     url = {http://www.numdam.org/articles/10.1051/ps:2007002/}
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Lepeltier, Jean-Pierre; Xu, Mingyu. Reflected backward stochastic differential equations with two RCLL barriers. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 3-22. doi : 10.1051/ps:2007002. http://www.numdam.org/articles/10.1051/ps:2007002/

[1] M. Alario-Nazaret, Jeux de Dynkin. Ph.D. dissertation, Univ. Franche-Comté, Besançon (1982).

[2] M. Alario-Nazaret, J.P. Lepeltier and B. Marchal, Dynkin games. Lect. Notes Control Inform. Sci. 43 (1982) 23-42.

[3] J.M. Bismut, Sur un problème de Dynkin. Z.Wahrsch. Verw. Gebiete 39 (1977) 31-53. | Zbl

[4] J. Cvitanic and I. Karatzas, Backward Stochastic Differential Equations with Reflection and Dynkin Games. Ann. Probab. 24 (1996) 2024-2056. | Zbl

[5] N. El Karoui, Les aspects probabilistes du contrôle stochastique, in P.L. Hennequin Ed., Ecole d'été de Saint-Flour. Lect. Notes Math. 876 (1979) 73-238. | Zbl

[6] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M.C. Quenez, Reflected Solutions of Backward SDE and Related Obstacle Problems for PDEs. Ann. Probab. 25 (1997) 702-737. | Zbl

[7] S. Hamadène, Reflected BSDE's with Discontinuous Barrier and Application. Stochastics and Stochastic Reports 74 (2002) 571-596. | Zbl

[8] J.P. Lepeltier and J. San Martín, Backward SDE's with two barriers and continuous coefficient. An existence result. J. Appl. Probab. 41 (2004) 162-175. | Zbl

[9] J.P. Lepeltier and M. Xu, Penalization method for Reflected Backward Stochastic Differential Equations with one RCLL barrier. Statistics Probab. Lett. 75 (2005) 58-66. | Zbl

[10] E. Pardoux and S. Peng, Adapted solutions of Backward Stochastic Differential Equations. Systems Control Lett. 14 (1990) 51-61. | Zbl

[11] S. Peng and M. Xu, Smallest g-Supermartingales and related Reflected BSDEs. Annales of I.H.P. 41 (2005) 605-630. | Numdam | Zbl

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