ArticleOriginal scientific text

Title

Discrete approximations of generalized RBSDE with random terminal time

Authors 1

Affiliations

  1. Institute of Mathematics and Physics, University of Technology and Life Sciences, Kaliskiego 7, 85-796 Bydgoszcz, Poland

Abstract

The convergence of discrete approximations of generalized reflected backward stochastic differential equations with random terminal time in a general convex domain is studied. Applications to investigation obstacle elliptic problem with Neumann boundary condition for partial differential equations are given.

Keywords

generalized reflected BSDE, discrete approximation methods, viscosity solution

Bibliography

  1. V. Bally, Approximation scheme for solutions of BSDE, Pitman Res. Notes Math. Ser. 364 Longman Harlow (1997) 177-191.
  2. V. Bally and G. Pages, Error analysis of the optimal quantization algorithm for obstacle problems, Stochastic Process. Appl. 106 (1) (2003) 1-40. doi: 10.1016/S0304-4149(03)00026-7.
  3. P. Briand, B. Delyon and J. Mémin, Donsker-Type theorem for BSDEs, Elect. Comm. in Probab. 6 (2001) 1-14.
  4. P. Briand, B. Delyon and J. Mémin, On the robustness of backward stochastic diffrential equations, Stochastic Process. Appl. 97 (2002) 229-253. doi: 10.1016/S0304-4149(01)00131-4.
  5. A. Gegout-Petit and É. Pardoux, Equations diffréntielles stochastiques rétrogrades réfléchies dans un convexe, Stoch. Stoch. Rep. 57 (1996) 111-128.
  6. K. Jańczak, Discrete approximations of reflected backward stochastic differential equations with random terminal time, Probab. Math. Statistics 28 (2008) 41-74.
  7. K. Jańczak, Generalized reflected backward stochastic differential equations, Stochastics 81 (2009) 147-170. doi: 10.1080/17442500802299007.
  8. K. Jańczak-Borkowska, Generalized RBSDE with random terminal time, Bull. Polish Acad. Sci. Math. 59 (2011) 85-100. doi: 10.4064/ba59-1-10.
  9. P.L. Lions and A.S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. (1984) 511-537.
  10. J. Ma, P. Protter, J. San Martin and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab. 12 (2002) 302-316. doi: 10.1214/aoap/1015961165.
  11. J. Ma and J. Zhang, Representation and regularities for solutions to BSDEs with reflections, Stochastic Process. Appl. 115 (2005) 539-569. doi: 10.1016/j.spa.2004.05.010.
  12. J.L. Menaldi, Stochastic variational inequality for reflected diffusion, Indiana Univ. Math. Journal 32 (1983) 733-744. doi: 10.1512/iumj.1983.32.32048.
  13. É. Pardoux and S. Peng, Adapted solutions of a backward stochastic differential equation, Systems Control Lett. 14 (1990) 55-61. doi: 10.1016/0167-6911(90)90082-6.
  14. É. Pardoux and A. Răşcanu, Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl. 76 (1998) 191-215. doi: 10.1016/S0304-4149(98)00030-1.
  15. É. Pardoux and S. Zhang, Generalized BSDEs and nonlinear Neumann boundary value problems, Probab. Theory Relat. Fields 110 (1998) 535-558. doi: 10.1007/s004400050158.
  16. Y. Ren and N. Xia, Generalized reflected BSDE and an obstacle problem for PDEs with a nonlinear Neumann boundary condition, Stochastic Analysis and Applications 24 (2006) 1-21. doi: 10.1080/07362990600870454.
  17. L. Słomiński, Euler's approximations of solutions of SDE's with reflecting boundary, Stochastic Process. Appl. 94 (2001) 317-337. doi: 10.1016/S0304-4149(01)00087-4.
  18. S. Toldo, Stability of solutions of BSDEs with random terminal time, SAIM Probab. Stat. 10 (2006) 141-163. doi: 10.1051/ps:2006006.
Pages:
69-85
Main language of publication
English
Received
2012-09-16
Published
2012
Exact and natural sciences