Reflected backward stochastic differential equations with two RCLL barriers

Jean-Pierre Lepeltier; Mingyu Xu

ESAIM: Probability and Statistics (2007)

  • Volume: 11, page 3-22
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Lepeltier, Jean-Pierre, and Xu, Mingyu. "Reflected backward stochastic differential equations with two RCLL barriers." ESAIM: Probability and Statistics 11 (2007): 3-22. <http://eudml.org/doc/250098>.

@article{Lepeltier2007,
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. },
author = {Lepeltier, Jean-Pierre, Xu, Mingyu},
journal = {ESAIM: Probability and Statistics},
keywords = {Reflected backward stochastic differential equation; penalization method; optimal stopping; Snell envelope; Dynkin game.; reflected backward stochastic differential equation; snell envelope; dynkin game},
language = {eng},
month = {3},
pages = {3-22},
publisher = {EDP Sciences},
title = {Reflected backward stochastic differential equations with two RCLL barriers},
url = {http://eudml.org/doc/250098},
volume = {11},
year = {2007},
}

TY - JOUR
AU - Lepeltier, Jean-Pierre
AU - Xu, Mingyu
TI - Reflected backward stochastic differential equations with two RCLL barriers
JO - ESAIM: Probability and Statistics
DA - 2007/3//
PB - EDP Sciences
VL - 11
SP - 3
EP - 22
AB - 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.
LA - eng
KW - Reflected backward stochastic differential equation; penalization method; optimal stopping; Snell envelope; Dynkin game.; reflected backward stochastic differential equation; snell envelope; dynkin game
UR - http://eudml.org/doc/250098
ER -

References

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  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. Gebiete39 (1977) 31–53.  
  4. J. Cvitanic and I. Karatzas, Backward Stochastic Differential Equations with Reflection and Dynkin Games. Ann. Probab.24 (1996) 2024–2056.  
  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.  
  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.  
  7. S. Hamadène, Reflected BSDE's with Discontinuous Barrier and Application. Stochastics and Stochastic Reports74 (2002) 571–596.  
  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.  
  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.  
  10. E. Pardoux and S. Peng, Adapted solutions of Backward Stochastic Differential Equations. Systems Control Lett.14 (1990) 51–61.  
  11. S. Peng and M. Xu, Smallestg-Supermartingales and related Reflected BSDEs. Annales of I.H.P.41 (2005) 605–630.  

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