Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations***

Shige Peng; Mingyu Xu

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 2, page 335-360
  • ISSN: 0764-583X

Abstract

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In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.

How to cite

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Peng, Shige, and Xu, Mingyu. "Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations***." ESAIM: Mathematical Modelling and Numerical Analysis 45.2 (2011): 335-360. <http://eudml.org/doc/276348>.

@article{Peng2011,
abstract = { In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs. },
author = {Peng, Shige, Xu, Mingyu},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Backward stochastic differential equations; reflected stochastic differential equations with one barrier; numerical algorithm; numerical simulation; backward stochastic differential equations; algorithm; numerical examples; Brownian motion; convergence},
language = {eng},
month = {1},
number = {2},
pages = {335-360},
publisher = {EDP Sciences},
title = {Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations***},
url = {http://eudml.org/doc/276348},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Peng, Shige
AU - Xu, Mingyu
TI - Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations***
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/1//
PB - EDP Sciences
VL - 45
IS - 2
SP - 335
EP - 360
AB - In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.
LA - eng
KW - Backward stochastic differential equations; reflected stochastic differential equations with one barrier; numerical algorithm; numerical simulation; backward stochastic differential equations; algorithm; numerical examples; Brownian motion; convergence
UR - http://eudml.org/doc/276348
ER -

References

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  1. V. Bally, An approximation scheme for BSDEs and applications to control and nonlinear PDE's, in Pitman Research Notes in Mathematics Series364, Longman, New York (1997).  
  2. V. Bally and G. Pages, A quantization algorithm for solving discrete time multi-dimensional optimal stopping problems. Bernoulli9 (2003) 1003–1049.  
  3. V. Bally and G. Pages, Error analysis of the quantization algorithm for obstacle problems. Stoch. Proc. Appl.106 (2003) 1–40.  
  4. B. Bouchard and N. Touzi, Discrete time approximation and Monte-Carlo simulation of backward stochastic differential equation. Stoch. Proc. Appl.111 (2004) 175–206.  
  5. P. Briand, B. Delyon and J. Mémin, Donsker-type theorem for BSDEs. Elect. Comm. Probab.6 (2001) 1–14.  
  6. P. Briand, B. Delyon and J. Mémin, On the robustness of backward stochastic differential equations. Stoch. Process. Appl.97 (2002) 229–253.  
  7. D. Chevance, Résolution numérique des équations différentielles stochastiques rétrogrades, in Numerical Methods in Finance, Cambridge University Press, Cambridge (1997).  
  8. F. Coquet, V. Mackevicius and J. Mémin, Stability in D of martingales and backward equations under discretization of filtration. Stoch. Process. Appl.75 (1998) 235–248.  
  9. J. Cvitanic, I. Karatzas and M. Soner, Backward stochastic differential equations with constraints on the gain-process. Ann. Probab.26 (1998) 1522–1551.  
  10. F. Delarue and S. Menozzi, An interpolated Stochastic Algorithm for Quasi-Linear PDEs. Math. Comput.261 (2008) 125–158.  
  11. J. Douglas, J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations. Ann. Appl. Probab.6 (1996) 940–968.  
  12. N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M.-C. Quenez, Reflected solutions of backward SDE and related obstacle problems for PDEs. Ann. Probab.25 (1997) 702–737.  
  13. N. El Karoui, S. Peng and M.C. Quenez, Backward stochastic differential equations in finance. Math. Finance7 (1997) 1–71.  
  14. E. Gobet, J.P. Lemor and X. Warin, Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli12 (2006) 889–916.  
  15. P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992).  
  16. J. Ma, P. Protter, J. San Martín and S. Torres, Numerical method for backward stochastic differential equations. Ann. Appl. Probab.12 (2002) 302–316.  
  17. J. Mémin, S. Peng and M. Xu, Convergence of solutions of discrete reflected backward SDE's and simulations. Acta Math. Appl. Sin. (English Series)24 (2008) 1–18.  
  18. E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation. Syst. Control Lett.14 (1990) 55–61.  
  19. S. Peng, Monotonic limit theory of BSDE and nonlinear decomposition theorem of Doob-Meyer's type. Probab. Theory Relat. Fields113 (1999) 473–499.  
  20. S. Peng and M. Xu, Reflected BSDE with Constraints and the Related Nonlinear Doob-Meyer Decomposition. Preprint, available at e-print:v4 (2006).  URIarXiv:math/0611869
  21. E.G. Rosazza, Risk measures via -expectations. Insur. Math. Econ.39 (2006) 19–34.  
  22. M. Xu, Numerical algorithms and simulations for reflected BSDE with two barriers. Preprint, available at arXiv:0803.3712v2 [math.PR] (2007).  
  23. J. Zhang, Some fine properties of backward stochastic differential equations. Ph.D. Thesis, Purdue University (2001).  
  24. J. Zhang, A numerical scheme for BSDEs. Ann. Appl. Probab.14 (2004) 459–488.  
  25. Y. Zhang and W. Zheng, Discretizing a backward stochastic differential equation. Int. J. Math. Math. Sci.32 (2002) 103–116.  

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