1. Home
  2. Issues
  3. Volume 4, Issue 4 (2017)
  4. Double barrier reflected BSDEs with stoc ...

Modern Stochastics: Theory and Applications


Double barrier reflected BSDEs with stochastic Lipschitz coefficient
Volume 4, Issue 4 (2017), pp. 353–379
Pub. online: 8 December 2017      Type: Research Article      Open accessOpen Access
Received
21 July 2017
Revised
14 November 2017
Accepted
14 November 2017
Published
8 December 2017

Abstract

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.

References

[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. http://hdl.handle.net/10419/85163
[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. https://arxiv.org/abs/0912.2162
[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)

Full article Related articles Cited by PDF XML
Full article Related articles Cited by PDF XML

Copyright
© 2017 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
BSDE and reflected BSDE Stochastic Lipschitz coefficient

MSC2010
60H20 60H30 65C30

Metrics
since March 2018
1070

Article info
views

690

Full article
views

480

PDF
downloads

161

XML
downloads


Share


RSS