Double barrier reflected BSDEs with stochastic Lipschitz coefficient

Mohamed Marzougue, Mohamed El Otmani


This paper proves the existence and uniqueness of a solution to doubly reflected backward stochastic differential equations where the coefficient is stochastic Lipschitz, by means of the penalization method.


BSDE and reflected BSDE; Stochastic Lipschitz coefficient

Full Text:



[1] Bahlali, K., Hamadène, S., Mezerdi, B.: Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient. Stochastic Processes and their Applications 115, 1107–1129 (2005). MR2147243

[2] Bender, C., Kohlmann, M.: BSDEs with Stochastic Lipschitz Condition.

[3] Bismut, J.M.: Conjugate convex functions in optimal stochastic control. Journal of Mathematical Analysis and Applications 44(2), 384–40 (1973). MR0329726

[4] Chung, K.: A Coure in Probability Theory. Academic Press (2001)

[5] Crépey, S., Matoussi, A.: Reflected and doubly reflected bsdes with jumps: a priori estimates and comparison. Ann. Appl. Probab 18, 2041–2069 (2008)

[6] Cvitanic, J., Karatzas, I.: Backward stochastic differential equations with reflection and dynkin games. The Annals of Probability 24(4), 2024–2056 (1996)

[7] Dellacherie, C., Meyer, P.: Probabilités et Potentiel I-IV. Hermann, Paris (1975). MR0488194

[8] El Karoui, N., Huang, S.J.: A general result of existence and uniqueness of backward stochastic differential equations. In: Pitman-Res-Notes-Math-Ser (ed.) Backward Stochastic Differential Equations. Springer, vol. 364, pp. 27–36 (1997)

[9] El Karoui, N., Quenez, M.C.: Non-linear pricing theory and backward stochastic differential equations. Financial mathematics 1656, 191–246 (1997)

[10] El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Mathematical Finance 7(1), 1–71 (1997)

[11] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M.C.: Reflected solutions of backward sde’s and related obstacle problems for pde’s. The Annals of Probability 25, 702–737 (1997). MR1434123

[12] El Otmani, M.: Approximation scheme for solutions of bsdes with two reflecting barriers. Stochastic Analysis and Applications 26(1), 60–83 (2008)

[13] El Otmani, M., Mrhardy, N.: Generalized bsde with two reflecting barriers. Random Operators and Stochastic Equations 16(4), 357–382 (2008)

[14] Essaky, E., Ouknine, Y., Harraj, N.: Backward stochastic differential equation with two reflecting barriers and jumps. Stochastic Analysis and Applications 23, 921–938 (2005). MR2158885

[15] Hamadène, S., Hassani, M.: Bsdes with two reflecting barriers : the general result. Probability Theory and Related Fields 132(2), 237–264 (2005)

[16] Hamadène, S., Hdhiri, I.: Backward stochastic differential equations with two distinct reflecting barriers and quadratic growth generator. J. Appl. Math. Stoch. Anal 2006, 1–28 (2006)

[17] Hamadène, S., Lepeltier, J.P.: Zero-sum stochastic differential games and backward equations. Systems and Control Letters 24(4), 259–263 (1995). MR1321134

[18] Lepeltier, J.P., San Martin, J.: Backward sdes with two barriers and continuous coefficient: an existence result. Journal of applied probability 41(1), 162–175 (2004)

[19] Li, M., Shi, Y.: Solving the double barriers reflected bsdes via penalization method. Statistics and Probability Letters 110, 74–83 (2016)

[20] Pardoux, E.: Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic pdes of second order. Stochastic Analysis and Related Topics VI 42, 79–127 (1998)

[21] Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Systems and Control Letters 1, 55–61 (1990)

[22] Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic partial differential Equations and their applications 176, 200–217 (1992)

[23] Protter, P.: Stochastic Integration and Differential Equations. Springer, Berlin (1990)

[24] Ren, Y., El Otmani, M.: Doubly reflected bsdes driven by a levy process. Nonlinear Analysis : Real World Applications 13, 1252–1267 (2012)

[25] Saisho, Y.: Stochastic differential equations for multi-dimensional domain with reflecting boundary. Probab. Theory Related Fields 74, 455–477 (1987)

[26] Wen, L.: Reflected BSDE with stochastic Lipschitz coefficient.

[27] Xu, M.: Reflected backward sdes with two barriers under monotonicity and general increasing conditions. Journal of Theoretical Probability 20(4), 1005–1039 (2007)

[28] Yosida, K.: Functional Analysis. Springer, New York (1980)

[29] Zheng, S., Zhou, S.: A generalized existence theorem of reflected bsdes with double obstacles. Statistics and Probability Letters 78(5), 528–536 (2007)



  • There are currently no refbacks.

Copyright (c) 2017 Mohamed Marzougue, Mohamed El Otmani

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.