Existence and uniqueness for \mathbb{D}-solutions of reflected BSDEs with two barriers without Mokobodzki's condition

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July 2016, 15(4): 1139-1156. doi: 10.3934/cpaa.2016.15.1139

Existence and uniqueness for $\mathbb{D}$-solutions of reflected BSDEs with two barriers without Mokobodzki's condition

Imen Hassairi ^1,

1.	Université de Sfax, Faculté des Sciences de Sfax, département de mathématiques, BP 1171 Sfax 3000, Tunisia

Received January 2015 Revised February 2016 Published April 2016

Abstract
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In this paper, we are interested in the problem of existence and uniqueness of a solution which belongs to class $\mathbb{D}$ for a backward stochastic differential equation with two strictly separated continuous reflecting barriers in the case when the data are $\mathbb{L}^1$-integrable and with generator satisfying the Lipschitz property. The main idea is to use the notion of local solution to obtain the global one.

Keywords: local solution, Doubly reflected BSDEs, Snell envelope., penalization, Mokobodzki's hypothesis.

Mathematics Subject Classification: Primary: 60G40, 60H20, 60F2.

Citation: Imen Hassairi. Existence and uniqueness for $\mathbb{D}$-solutions of reflected BSDEs with two barriers without Mokobodzki's condition. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1139-1156. doi: 10.3934/cpaa.2016.15.1139

References:

[1]	Ph. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of backward stochastic differential equations, Stochastic Process. Appl., 108 (2003), 109-129. doi: 10.1016/S0304-4149(03)00089-9.
[2]	J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Annals of probability, 24 (1996), 2024-2056. doi: 10.1214/aop/1041903216.
[3]	C. Dellacherie and P. A. Meyer, Probabilit et Potentiel I-IV, Hermann, Paris, 1975.
[4]	C. Dellacherie and P. A. Meyer, Probabilit et Potentiel V-VIII Hermann, Paris, 1980
[5]	B. El Asri, S. Hamade and H. Wang, $L^p$ solutions for doubly reflected backward stochastic differential equations, Stochastic Analysis and Applications, 29 (2011), 907-932. doi: 10.1080/07362994.2011.564442.
[6]	N. El Karoui, Les aspects probabilistes du contrôle stochastique, in Ecole dEtde Probabilit de Saint-Flour IX, Volume 876 of the series Lecture Notes in Mathematics , (1979), 73-238.
[7]	N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDEs, Ann Probab., 25 (1997), 702-737. doi: 10.1214/aop/1024404416.
[8]	S. Hamadène, Mixed zero-sum differential game and American game options, SIAM J. Control Optim., 45 (2006), 496-518. doi: 10.1137/S036301290444280X.
[9]	S. Hamadène, Reflected BSDE's with discontinuous barriers and application, Stochastics and Sotochastics Reports, (2002), 571-596.
[10]	S. Hamadène and M. Hassani, BSDEs with two reflecting barriers: the general result, Probability theory and related fields, 132 (2005), 237-264. doi: 10.1007/s00440-004-0395-2.
[11]	S. Hamadène and M. Jeanblanc, On the stopping and starting problem: application to reversible investment, Mathematics of Operations Research, 32 (2007), 182-192. doi: 10.1287/moor.1060.0228.
[12]	S. Hamadène and J. P. Lepeltier, Reflected backward SDE's and mixed game problems, Stochastic Processes and their Applications, 85 (2000), 177-188. doi: 10.1016/S0304-4149(99)00072-1.
[13]	S. Hamadène and Y. Ouknine, Reflected backward stochastic differential equation with jumps and random obstacle, Electronic Journal of Probability. 8 (2003), 1-20. doi: 10.1214/EJP.v8-124.
[14]	S. Hamadène and A. Popier, $L^p$ solutions for reflected backward stochastic differential equations, Stochastics and Dynamics, 12 (2012), 35 pages. doi: 10.1142/S0219493712003651.
[15]	T. Klimsiak, Reflected BSDEs with monotone generator, Electronic Journal of Probability, 107 (2012), 1-25. doi: 10.1214/EJP.v17-1759.
[16]	T. Klimsiak, BSDEs with monotone generator and two irregular reflecting barriers, Bulletin des Sciences Math閙atiques, 137 (2013), 268-321. doi: 10.1016/j.bulsci.2012.06.006.
[17]	E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equations, Systems and Control Letters, 14, (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.
[18]	S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Probability Theory and Related Fields, 113 (1999), 473-499. doi: 10.1007/s004400050214.
[19]	E. P. Protter, Stochastic Integration and Differential Equations, 2nd edition, Springer, New York, 2000.
[20]	D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, New York, 1994.
[21]	A. Roskoz and L. Slominski, $L^p$ solutions for reflected BSDEs under monotonicity condition, Stochastic Processes and their Applications, 122 (2012), 3875-3900. doi: 10.1016/j.spa.2012.07.006.

show all references

References:

[1]	Ph. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of backward stochastic differential equations, Stochastic Process. Appl., 108 (2003), 109-129. doi: 10.1016/S0304-4149(03)00089-9.
[2]	J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Annals of probability, 24 (1996), 2024-2056. doi: 10.1214/aop/1041903216.
[3]	C. Dellacherie and P. A. Meyer, Probabilit et Potentiel I-IV, Hermann, Paris, 1975.
[4]	C. Dellacherie and P. A. Meyer, Probabilit et Potentiel V-VIII Hermann, Paris, 1980
[5]	B. El Asri, S. Hamade and H. Wang, $L^p$ solutions for doubly reflected backward stochastic differential equations, Stochastic Analysis and Applications, 29 (2011), 907-932. doi: 10.1080/07362994.2011.564442.
[6]	N. El Karoui, Les aspects probabilistes du contrôle stochastique, in Ecole dEtde Probabilit de Saint-Flour IX, Volume 876 of the series Lecture Notes in Mathematics , (1979), 73-238.
[7]	N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDEs, Ann Probab., 25 (1997), 702-737. doi: 10.1214/aop/1024404416.
[8]	S. Hamadène, Mixed zero-sum differential game and American game options, SIAM J. Control Optim., 45 (2006), 496-518. doi: 10.1137/S036301290444280X.
[9]	S. Hamadène, Reflected BSDE's with discontinuous barriers and application, Stochastics and Sotochastics Reports, (2002), 571-596.
[10]	S. Hamadène and M. Hassani, BSDEs with two reflecting barriers: the general result, Probability theory and related fields, 132 (2005), 237-264. doi: 10.1007/s00440-004-0395-2.
[11]	S. Hamadène and M. Jeanblanc, On the stopping and starting problem: application to reversible investment, Mathematics of Operations Research, 32 (2007), 182-192. doi: 10.1287/moor.1060.0228.
[12]	S. Hamadène and J. P. Lepeltier, Reflected backward SDE's and mixed game problems, Stochastic Processes and their Applications, 85 (2000), 177-188. doi: 10.1016/S0304-4149(99)00072-1.
[13]	S. Hamadène and Y. Ouknine, Reflected backward stochastic differential equation with jumps and random obstacle, Electronic Journal of Probability. 8 (2003), 1-20. doi: 10.1214/EJP.v8-124.
[14]	S. Hamadène and A. Popier, $L^p$ solutions for reflected backward stochastic differential equations, Stochastics and Dynamics, 12 (2012), 35 pages. doi: 10.1142/S0219493712003651.
[15]	T. Klimsiak, Reflected BSDEs with monotone generator, Electronic Journal of Probability, 107 (2012), 1-25. doi: 10.1214/EJP.v17-1759.
[16]	T. Klimsiak, BSDEs with monotone generator and two irregular reflecting barriers, Bulletin des Sciences Math閙atiques, 137 (2013), 268-321. doi: 10.1016/j.bulsci.2012.06.006.
[17]	E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equations, Systems and Control Letters, 14, (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.
[18]	S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Probability Theory and Related Fields, 113 (1999), 473-499. doi: 10.1007/s004400050214.
[19]	E. P. Protter, Stochastic Integration and Differential Equations, 2nd edition, Springer, New York, 2000.
[20]	D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, New York, 1994.
[21]	A. Roskoz and L. Slominski, $L^p$ solutions for reflected BSDEs under monotonicity condition, Stochastic Processes and their Applications, 122 (2012), 3875-3900. doi: 10.1016/j.spa.2012.07.006.

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