Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T17:00:37.662Z Has data issue: false hasContentIssue false
  • English
  • Français

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: nielf0614@126.com
*Corresponding author.Email address:wdzhao@sdu.edu.cn
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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