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

Published online by Cambridge University Press:  24 August 2010

Shige Peng  and
Mingyu Xu
Shige Peng
Affiliation:
School of Mathematics and System Science, Shandong University, 250100 Jinan, P.R. China. peng@sdu.edu.cn. Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, CAS (No. 2008DP173182), P.R. China. xumy@amss.ac.cn.
Mingyu Xu
Affiliation:
Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, CAS (No. 2008DP173182), P.R. China. xumy@amss.ac.cn. Department of Financial Mathematics and Control science, School of Mathematical Science, Fudan University, 200433 Shanghai, P.R. China.
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Abstract

In this paper we study different algorithms for backwardstochastic differential equations (BSDE in short) basing on randomwalk framework for 1-dimensional Brownian motion. Implicit andexplicit schemes for both BSDE and reflected BSDE are introduced.Then we prove the convergence of different algorithms and presentsimulation results for different types of BSDEs.

Keywords

Backward stochastic differentialequationsreflected stochastic differential equations with one barriernumerical algorithmnumerical simulation

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 2 , March 2011 , pp. 335 - 360
Copyright
© EDP Sciences, SMAI, 2010

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References

V. Bally, An approximation scheme for BSDEs and applications to control and nonlinear PDE's, in Pitman Research Notes in Mathematics Series 364, Longman, New York (1997).
Bally, V. and Pages, G., A quantization algorithm for solving discrete time multi-dimensional optimal stopping problems. Bernoulli 9 (2003) 1003–1049. CrossRef
Bally, V. and Pages, G., Error analysis of the quantization algorithm for obstacle problems. Stoch. Proc. Appl. 106 (2003) 1–40. CrossRef
Bouchard, B. and Touzi, N., Discrete time approximation and Monte-Carlo simulation of backward stochastic differential equation. Stoch. Proc. Appl. 111 (2004) 175–206. CrossRef
Briand, P., Delyon, B. and Mémin, J., Donsker-type theorem for BSDEs. Elect. Comm. Probab. 6 (2001) 1–14. CrossRef
Briand, P., Delyon, B. and Mémin, J., On the robustness of backward stochastic differential equations. Stoch. Process. Appl. 97 (2002) 229–253. CrossRef
D. Chevance, Résolution numérique des équations différentielles stochastiques rétrogrades, in Numerical Methods in Finance, Cambridge University Press, Cambridge (1997).
Coquet, F., Mackevicius, V. and Mémin, J., Stability in D of martingales and backward equations under discretization of filtration. Stoch. Process. Appl. 75 (1998) 235–248. CrossRef
Cvitanic, J., Karatzas, I. and Soner, M., Backward stochastic differential equations with constraints on the gain-process. Ann. Probab. 26 (1998) 1522–1551.
Delarue, F. and Menozzi, S., An interpolated Stochastic Algorithm for Quasi-Linear PDEs. Math. Comput. 261 (2008) 125–158. CrossRef
Douglas, J., Ma, J. and Protter, P., Numerical methods for forward-backward stochastic differential equations. Ann. Appl. Probab. 6 (1996) 940–968.
El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M.-C., Reflected solutions of backward SDE and related obstacle problems for PDEs. Ann. Probab. 25 (1997) 702–737.
El Karoui, N., Peng, S. and Quenez, M.C., Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1–71. CrossRef
Gobet, E., Lemor, J.P. and Warin, X., Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli 12 (2006) 889–916.
P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992).
Ma, J., Protter, P., San Martín, J. and Torres, S., Numerical method for backward stochastic differential equations. Ann. Appl. Probab. 12 (2002) 302–316.
Mémin, J., Peng, S. and Convergence, M. Xu of solutions of discrete reflected backward SDE's and simulations. Acta Math. Appl. Sin. (English Series) 24 (2008) 1–18. CrossRef
Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14 (1990) 55–61. CrossRef
Peng, S., Monotonic limit theory of BSDE and nonlinear decomposition theorem of Doob-Meyer's type. Probab. Theory Relat. Fields 113 (1999) 473–499. CrossRef
S. Peng and M. Xu, Reflected BSDE with Constraints and the Related Nonlinear Doob-Meyer Decomposition. Preprint, available at e-print:arXiv:math/0611869v4 (2006).
Rosazza, E.G., Risk measures via $g$ -expectations. Insur. Math. Econ. 39 (2006) 19–34. CrossRef
M. Xu, Numerical algorithms and simulations for reflected BSDE with two barriers. Preprint, available at arXiv:0803.3712v2 [math.PR] (2007).
J. Zhang, Some fine properties of backward stochastic differential equations. Ph.D. Thesis, Purdue University (2001).
Zhang, J., A numerical scheme for BSDEs. Ann. Appl. Probab. 14 (2004) 459–488. CrossRef
Zhang, Y. and Zheng, W., Discretizing a backward stochastic differential equation. Int. J. Math. Math. Sci. 32 (2002) 103–116. CrossRef