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An Explicit Second-Order Numerical Scheme to Solve Decoupled Forward Backward Stochastic Equations

Published online by Cambridge University Press:  16 July 2018

Yu Fu  and
Weidong Zhao
Yu Fu*
Affiliation:
School of Mathematics & Finance Institute, Shandong University, Jinan 250100, China.
Weidong Zhao*
Affiliation:
School of Mathematics & Finance Institute, Shandong University, Jinan 250100, China.
*
Corresponding author.Email address: [email protected]
*Corresponding author.Email address:[email protected]
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Abstract

An explicit numerical scheme is proposed for solving decoupled forward backward stochastic differential equations (FBSDE) represented in integral equation form. A general error inequality is derived for this numerical scheme, which also implies its stability. Error estimates are given based on this inequality, showing that the explicit scheme can be second-order. Some numerical experiments are carried out to illustrate the high accuracy of the proposed scheme.

Keywords

60H3565C2060H10Explicit schemesecond-orderdecoupled FBSDEerror estimate
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 4 , November 2014 , pp. 368 - 385
Copyright
Copyright © Global-Science Press 2014

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References

[1] Bender, C. and Denk, R., A forward scheme for backward sdes, Stoch. Proc. Appl. 117, 1793–1812 (2007).Google Scholar
[2] Delarue, F., On the existence and uniqueness of solutions to fbsdes in a non-degenerate case, Stoch. Proc. Appl. 99, 209–286 (2002.Google Scholar
[3] Douglas, J., Ma, J. and Protter, P., Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab. 6, 940–968 (1996).Google Scholar
[4] Gobet, E. and Labart, C., Error expansion for the discretization of backward stochastic differential equations, Stoch. Proc. Appl. 117, 803–829 (2007).CrossRefGoogle Scholar
[5] Karou, N. NL., Peng, S., and Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance 7, 1–71 (1997).CrossRefGoogle Scholar
[6] Kloeden, P. E. and Platen, E., Numerical Solutions of Stochastic Differential Equations, Springer, Berlin and Heidelberg (1992).Google Scholar
[7] Ma, J., Protter, P., Martin, J. S., and Torres, S., Numerical method for backward stochastic differential equations, Ann. Appl. Probab. 12, 302–316 (2002).Google Scholar
[8] Ma, J., Protter, P., and Yong, J., Solving forward-backward stochastic differential equations explicitly - a four step scheme, Probab. Theory Related Fields 98, 339–359 (1994).Google Scholar
[9] Ma, J., Shen, J., and Zhao, Y., On numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal. 46, 2636–2661 (2008).Google Scholar
[10] Ma, J. and Yong, J., Forward-Backward Stochastic Differential Equations and their Applications, Springer, Berlin (1999).Google Scholar
[11] Milstein, G. N. and Tretyakov, M. V., Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput. 28, 561–582 (2006).Google Scholar
[12] Oksendal, B., Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin (2003).CrossRefGoogle Scholar
[13] Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14, 56–61 (1990).Google Scholar
[14] Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastic Reports 37, 61–74 (1991).Google Scholar
[15] Zhao, W., Chen, L., and Peng, S., A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput. 28, 1563–1581 (2006).Google Scholar
[16] 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, A1731–A1751 (2014).Google Scholar
[17] Zhao, W., Zhang, G., and Ju, L., A stable multistep schemes for backward stochastic differential equations, SIAM J. Numer. Anal. 48, 1369–1394 (2010).Google Scholar
[18] Zhao, W., Zhang, W., and Ju, L., A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations, Comm. Comput. Phys. 15, 618–646 (2014).Google Scholar