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Deferred Correction Methods for Forward Backward Stochastic Differential Equations

Part of: Stochastic analysis Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  09 May 2017

Tao Tang ,
Weidong Zhao  and
Tao Zhou
Tao Tang*
Affiliation:
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China; and Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Weidong Zhao*
Affiliation:
School of Mathematics & Finance Institute, Shandong University, Jinan 250100, China
Tao Zhou*
Affiliation:
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
*Corresponding author. Email addresses:tangt@sustc.edu.cn (T. Tang), wdzhao@sdu.edu.cn (W. D. Zhao), tzhou@lsec.cc.ac.cn (T. Zhou)
*Corresponding author. Email addresses:tangt@sustc.edu.cn (T. Tang), wdzhao@sdu.edu.cn (W. D. Zhao), tzhou@lsec.cc.ac.cn (T. Zhou)
*Corresponding author. Email addresses:tangt@sustc.edu.cn (T. Tang), wdzhao@sdu.edu.cn (W. D. Zhao), tzhou@lsec.cc.ac.cn (T. Zhou)
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Abstract

The deferred correction (DC) method is a classical method for solving ordinary differential equations; one of its key features is to iteratively use lower order numerical methods so that high-order numerical scheme can be obtained. The main advantage of the DC approach is its simplicity and robustness. In this paper, the DC idea will be adopted to solve forward backward stochastic differential equations (FBSDEs) which have practical importance in many applications. Noted that it is difficult to design high-order and relatively “clean” numerical schemes for FBSDEs due to the involvement of randomness and the coupling of the FSDEs and BSDEs. This paper will describe how to use the simplest Euler method in each DC step–leading to simple computational complexity–to achieve high order rate of convergence.

Keywords

Deferred correction methodforward backward stochastic differential equationsEuler methodhigh-order scheme

MSC classification

Secondary: 60H35: Computational methods for stochastic equations 65C20: Models, numerical methods 60H10: Stochastic ordinary differential equations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017 , pp. 222 - 242
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Bally, V., Approximation scheme for solutions of BSDE, in Backward Stochastic Differential Equations, Pitman Res. Notes Math. 364, Longman, Harlow, UK, 1997, pp. 177–191.Google Scholar
[2] Bender, C. and Denk, R., A forward scheme for backward SDEs, Stochastic Process. Appl., 117 (2007), pp. 1793–1812.Google Scholar
[3] Bouchard, B. and Touzi, N., Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 111 (2004), pp. 175–206.CrossRefGoogle Scholar
[4] Bourlioux, A., Layton, A.T., Minion, M.L., High-order multi-implicit spectral deferre correction methods for problems of reactive flow, J. Comput. Phys., 189 (2003), pp. 351–376.Google Scholar
[5] Dutt, A., Greengard, L. and Rokhlin, V., Spectral deferred correction methods for ordinary differential equations, BIT, 40 (2000), pp. 241–266.Google Scholar
[6] Douglas, J., Ma, J. and Protter, P., Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6(1996), pp. 940–968.Google Scholar
[7] Feng, X., Tang, T. and Yang, J., Long time numerical simulations for phase-field problems using p-adaptive spectral deferred correction methods, SIAM J. Sci. Comput., 37 (2015), A2710–A294.Google Scholar
[8] Fu, Y., Zhao, W. and Zhou, T., Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs, arXiv:1607.06897, submitted, 2016.Google Scholar
[9] Hairer, E., On the order of iterated defect correction, Numer. Math. 29 (1978), pp. 409–424.Google Scholar
[10] Hansen, A. and Strain, J., On the order of deferred correction, Appl. Numer. Math., 61 (2011), pp. 961–973.Google Scholar
[11] Milstein, G. N. and Tretyakov, M. V., Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), pp. 561–582.Google Scholar
[12] Pardoux, E. and Peng, S., Adapted solution of a backword stochastic differential equaion, Systems Control Lett., 14 (1990), pp. 55–61.Google Scholar
[13] Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Reps., 37 (1991), pp. 61–74.Google Scholar
[14] Tang, T., Xie, H. and Yin, X., High-order convergence of spectral deferred correction method on general quadrature nodes, J. Sci. Comput., 56 (2013), pp. 1–13.Google Scholar
[15] Pereyra, V., Iterated deferred correction for nonlinear boundary value problems, Numer. Math., 11 (1968), pp. 111–125.Google Scholar
[16] Zadunaisky, P., A method for the estimation of errors propagated in the numerical solution of a system of ordinary differential equations, The Theory of Orbits in the Solar System and in Stellar Systems, in: Proceedings on International Astronomical Union, Symposium 25, 1964.Google Scholar
[17] Zhao, W., Chen, L. and Peng, S., A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), pp. 1563–1581.Google Scholar
[18] Zhao, W., Fu, Y. and Zhou, T., New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput., 36(4) (2014), pp. A1731–1751.Google Scholar
[19] Zhao, W., Li, Y. and Zhang, G., A generalized θ-scheme for solving backward stochastic differential equations, Discrete Contin. Dynam. Systems. Ser. B, 17 (2012), pp. 1585–1603.Google Scholar
[20] Zhao, W., Zhang, G. and Ju, L., A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), pp. 1369–1394.CrossRefGoogle Scholar