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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

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 & 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,, \emph{Stochastic Process. Appl.}, 108 (2003), 109.  doi: 10.1016/S0304-4149(03)00089-9.  Google Scholar

[2]

J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games,, \emph{Annals of probability}, 24 (1996), 2024.  doi: 10.1214/aop/1041903216.  Google Scholar

[3]

C. Dellacherie and P. A. Meyer, Probabilit et Potentiel I-IV,, Hermann, (1975).   Google Scholar

[4]

C. Dellacherie and P. A. Meyer, Probabilit et Potentiel V-VIII, Hermann, (1980).   Google Scholar

[5]

B. El Asri, S. Hamade and H. Wang, $L^p$ solutions for doubly reflected backward stochastic differential equations,, \emph{Stochastic Analysis and Applications}, 29 (2011), 907.  doi: 10.1080/07362994.2011.564442.  Google Scholar

[6]

N. El Karoui, Les aspects probabilistes du contrôle stochastique,, in \emph{ Ecole dEtde Probabilit de Saint-Flour IX, (1979), 73.   Google Scholar

[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,, \emph{Ann Probab.}, 25 (1997), 702.  doi: 10.1214/aop/1024404416.  Google Scholar

[8]

S. Hamadène, Mixed zero-sum differential game and American game options,, \emph{SIAM J. Control Optim.}, 45 (2006), 496.  doi: 10.1137/S036301290444280X.  Google Scholar

[9]

S. Hamadène, Reflected BSDE's with discontinuous barriers and application,, \emph{Stochastics and Sotochastics Reports}, (2002), 571.   Google Scholar

[10]

S. Hamadène and M. Hassani, BSDEs with two reflecting barriers: the general result,, \emph{Probability theory and related fields}, 132 (2005), 237.  doi: 10.1007/s00440-004-0395-2.  Google Scholar

[11]

S. Hamadène and M. Jeanblanc, On the stopping and starting problem: application to reversible investment,, \emph{Mathematics of Operations Research}, 32 (2007), 182.  doi: 10.1287/moor.1060.0228.  Google Scholar

[12]

S. Hamadène and J. P. Lepeltier, Reflected backward SDE's and mixed game problems,, \emph{Stochastic Processes and their Applications}, 85 (2000), 177.  doi: 10.1016/S0304-4149(99)00072-1.  Google Scholar

[13]

S. Hamadène and Y. Ouknine, Reflected backward stochastic differential equation with jumps and random obstacle,, \emph{Electronic Journal of Probability}. \textbf{8} (2003), 8 (2003), 1.  doi: 10.1214/EJP.v8-124.  Google Scholar

[14]

S. Hamadène and A. Popier, $L^p$ solutions for reflected backward stochastic differential equations,, \emph{Stochastics and Dynamics}, 12 (2012).  doi: 10.1142/S0219493712003651.  Google Scholar

[15]

T. Klimsiak, Reflected BSDEs with monotone generator,, \emph{Electronic Journal of Probability}, 107 (2012), 1.  doi: 10.1214/EJP.v17-1759.  Google Scholar

[16]

T. Klimsiak, BSDEs with monotone generator and two irregular reflecting barriers,, \emph{Bulletin des Sciences Math閙atiques}, 137 (2013), 268.  doi: 10.1016/j.bulsci.2012.06.006.  Google Scholar

[17]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equations,, \emph{Systems and Control Letters}, 14 (1990), 55.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[18]

S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type,, \emph{Probability Theory and Related Fields}, 113 (1999), 473.  doi: 10.1007/s004400050214.  Google Scholar

[19]

E. P. Protter, Stochastic Integration and Differential Equations,, 2$^{nd}$ edition, (2000).   Google Scholar

[20]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion,, Springer, (1994).   Google Scholar

[21]

A. Roskoz and L. Slominski, $L^p$ solutions for reflected BSDEs under monotonicity condition,, \emph{Stochastic Processes and their Applications}, 122 (2012), 3875.  doi: 10.1016/j.spa.2012.07.006.  Google Scholar

show all references

References:
[1]

Ph. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of backward stochastic differential equations,, \emph{Stochastic Process. Appl.}, 108 (2003), 109.  doi: 10.1016/S0304-4149(03)00089-9.  Google Scholar

[2]

J. Cvitanic and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games,, \emph{Annals of probability}, 24 (1996), 2024.  doi: 10.1214/aop/1041903216.  Google Scholar

[3]

C. Dellacherie and P. A. Meyer, Probabilit et Potentiel I-IV,, Hermann, (1975).   Google Scholar

[4]

C. Dellacherie and P. A. Meyer, Probabilit et Potentiel V-VIII, Hermann, (1980).   Google Scholar

[5]

B. El Asri, S. Hamade and H. Wang, $L^p$ solutions for doubly reflected backward stochastic differential equations,, \emph{Stochastic Analysis and Applications}, 29 (2011), 907.  doi: 10.1080/07362994.2011.564442.  Google Scholar

[6]

N. El Karoui, Les aspects probabilistes du contrôle stochastique,, in \emph{ Ecole dEtde Probabilit de Saint-Flour IX, (1979), 73.   Google Scholar

[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,, \emph{Ann Probab.}, 25 (1997), 702.  doi: 10.1214/aop/1024404416.  Google Scholar

[8]

S. Hamadène, Mixed zero-sum differential game and American game options,, \emph{SIAM J. Control Optim.}, 45 (2006), 496.  doi: 10.1137/S036301290444280X.  Google Scholar

[9]

S. Hamadène, Reflected BSDE's with discontinuous barriers and application,, \emph{Stochastics and Sotochastics Reports}, (2002), 571.   Google Scholar

[10]

S. Hamadène and M. Hassani, BSDEs with two reflecting barriers: the general result,, \emph{Probability theory and related fields}, 132 (2005), 237.  doi: 10.1007/s00440-004-0395-2.  Google Scholar

[11]

S. Hamadène and M. Jeanblanc, On the stopping and starting problem: application to reversible investment,, \emph{Mathematics of Operations Research}, 32 (2007), 182.  doi: 10.1287/moor.1060.0228.  Google Scholar

[12]

S. Hamadène and J. P. Lepeltier, Reflected backward SDE's and mixed game problems,, \emph{Stochastic Processes and their Applications}, 85 (2000), 177.  doi: 10.1016/S0304-4149(99)00072-1.  Google Scholar

[13]

S. Hamadène and Y. Ouknine, Reflected backward stochastic differential equation with jumps and random obstacle,, \emph{Electronic Journal of Probability}. \textbf{8} (2003), 8 (2003), 1.  doi: 10.1214/EJP.v8-124.  Google Scholar

[14]

S. Hamadène and A. Popier, $L^p$ solutions for reflected backward stochastic differential equations,, \emph{Stochastics and Dynamics}, 12 (2012).  doi: 10.1142/S0219493712003651.  Google Scholar

[15]

T. Klimsiak, Reflected BSDEs with monotone generator,, \emph{Electronic Journal of Probability}, 107 (2012), 1.  doi: 10.1214/EJP.v17-1759.  Google Scholar

[16]

T. Klimsiak, BSDEs with monotone generator and two irregular reflecting barriers,, \emph{Bulletin des Sciences Math閙atiques}, 137 (2013), 268.  doi: 10.1016/j.bulsci.2012.06.006.  Google Scholar

[17]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equations,, \emph{Systems and Control Letters}, 14 (1990), 55.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[18]

S. Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type,, \emph{Probability Theory and Related Fields}, 113 (1999), 473.  doi: 10.1007/s004400050214.  Google Scholar

[19]

E. P. Protter, Stochastic Integration and Differential Equations,, 2$^{nd}$ edition, (2000).   Google Scholar

[20]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion,, Springer, (1994).   Google Scholar

[21]

A. Roskoz and L. Slominski, $L^p$ solutions for reflected BSDEs under monotonicity condition,, \emph{Stochastic Processes and their Applications}, 122 (2012), 3875.  doi: 10.1016/j.spa.2012.07.006.  Google Scholar

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