Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-16T17:26:30.388Z Has data issue: false hasContentIssue false
  • English
  • Français

A Numerical Method and its Error Estimates for the Decoupled Forward-Backward Stochastic Differential Equations

Published online by Cambridge University Press:  03 June 2015

Weidong Zhao ,
Wei Zhang  and
Lili Ju
Weidong Zhao*
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China
Wei Zhang*
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China
Lili Ju*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA Beijing Computational Science Research Center, Beijing 100084, P.R. China
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

In this paper, a new numerical method for solving the decoupled forward-backward stochastic differential equations (FBSDEs) is proposed based on some specially derived reference equations. We rigorously analyze errors of the proposed method under general situations. Then we present error estimates for each of the specific cases when some classical numerical schemes for solving the forward SDE are taken in the method; in particular, we prove that the proposed method is second-order accurate if used together with the order-2.0 weak Taylor scheme for the SDE. Some examples are also given to numerically demonstrate the accuracy of the proposed method and verify the theoretical results.

Keywords

60H3560H1065C2065C30Decoupled forward-backward stochastic differential equationsnumerical schemeerror estimates
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 3 , March 2014 , pp. 618 - 646
Copyright
Copyright © Global Science Press Limited 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]Bally, V. and Talay, D., The law of Euler scheme for stochastic differential equations, Probability Theroy and Related Fields, 104(1996), 43–60.Google Scholar
[2]Bender, C. and Denk, R., A forward scheme for backward SDEs, Stochastic Processes and their Applications, 117(2007), 1793–1812.Google Scholar
[3]Bouchard, B. and Elie, R., Discrete-time approximation of decoupled Forward-Backward SDE with jumps, Stochastic Processes and their Applications, 118(2008), 53–75.Google Scholar
[4]Bouchard, B. and Touzi, N., Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications, 111(2004), 175–206.Google Scholar
[5]Bouchard, B., Elie, R., and Touzi, N., Discrete-time approximation of BSDEs and probabilistic schemes for fully nonlinear PDEs, Randon Series Comp. Appl. Math., 8(2009), 1–34.Google Scholar
[6]Chevance, D., Numerical Methods for Backward Stochastic Differential Equations, in Numerical Methods in Finance, Edited by Rogers, L. C. G. and Talay, D., Cambridge University Press, 1997, 232–244.Google Scholar
[7]Crisan, D. and Manolarakis, K., Probabilistic methods for semilinear partial differential equations, ESAIM: Mathematical Modelling and Numerical Analysis, 44(2010), 1107–1133.CrossRefGoogle Scholar
[8]Cvitanic, J. and Zhang, J., The Steepest Descent Method for FBSDEs, Electronic Journal of Probability, 10(2005), 1468–1495.Google Scholar
[9]Delarue, F., On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stoachstic Process and their Applications, 99(2002), 209–286.Google Scholar
[10]Delarue, F. and Menozzi, S., A forward-backward stochastic algorithm for quasi-linear PDEs, The Annals of Applied Probability, 16(2006), 140–184.Google Scholar
[11]Douglas, J., Ma, J., and Protter, P., Numerical methods for forward-backward stochastic differential equations, The Annals of Applied Probability, 6(1996), 940–968.Google Scholar
[12]Evans, L., Partial Differential Equations, Providence, R.I. : American Mathematical Society, 1998.Google Scholar
[13]Gobet, E. and Labart, C., Error expansion for the discretization of Backward Stochastic Differential Equations, Stochastic Processes and their Applications, 117(2007), 803–829.Google Scholar
[14]Gobet, E., Lemmor, P., and Warin, X., A regression-based Monte Carlo method to solve backward stochastic differential equations, The Annals of Applied Probability, 151(2005), 2172–2202.Google Scholar
[15]Karoui, N., Peng, S., and Quenez, M., Backward stochastic differential equations in finance, Mathematical Finance, 7(1997), 1–71.CrossRefGoogle Scholar
[16]Kloeden, P. and Platen, E., Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, Third Printing, 1999.Google Scholar
[17]Ladyzenskaja, O., Solonnikov, O., and Uralceva, N., Linear and Quasi-Linear Equations of Parabolic Type, Translations of Math. Monographs, 23, AMS, Providence Rhode Island, 1968.Google Scholar
[18]Li, Y. and Zhao, W., Lp-error estimates for numerical schemes for solving certain kinds of backward stochastic differential equations, Statistics and Probability Letters, 80(2010), 1612–1617.CrossRefGoogle Scholar
[19]Ma, J., Protter, P., and Yong, J., Solving forward-backward stochastic differential equations explicitly-a four step scheme, Probability Theory and Related Fields, 98(1994), 339–359.Google Scholar
[20]Ma, J., Shen, J., and Zhao, Y., On Numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal., 46(2008), 2636–2661.Google Scholar
[21]Ma, J. and Yong, J., Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math., 1702, Springer, 1999.Google Scholar
[22]Ma, J. and Zhang, J., Representation theorems for backward stochastic differential equations, The Annals of Applied Probability, 12(2002), 1390–1418.Google Scholar
[23]Milstein, G. and Tretyakov, M. V., Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28(2006), 561–582.Google Scholar
[24]Milstein, G. and Tretyakov, M. V., Discretization of forward-backward stochastic differential equations and related quasi-linear parabolic equations, IMA J. Numer. Anal., 27(2007), 24–44.Google Scholar
[25]Pardoux, E. and Peng, S.Adapted solution of a backward stochastic differntial equation, Systems and Control Letters, 14(1990), 55–61.Google Scholar
[26]Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports, 37(1991), 61–74.Google Scholar
[27]Pardoux, E., Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, Stochastic analysis and related topics VI (Geilo, 1996), 79–127, Progr. Probab., 42, Birkhauser Boston, Boston, MA, 1998.Google Scholar
[28]Peng, S., A linear approximation algorithm using BSDE, Pacific Economic Review, 4(1999), 285–291.Google Scholar
[29]Zhang, J., A numerical scheme for BSDEs, The Annals of Applied Probability, 14(2004), 459–488.CrossRefGoogle Scholar
[30]Zhang, J., The wellposedness of FBSDEs, Dis. Con. Dyn. Sys. B, 6(2006), 927–940.Google Scholar
[31]Zhang, Y. and Zheng, W., Discretizing a backward stochastic differential equation, Inter. Mathematics, J. and Mathematical Sciences, 32(2002), 103–116.Google Scholar
[32]Zhao, W., Li, Y. and Zhang, G., A generalized θ-scheme for solving backward stochastic differential equations, Dis. Con. Dyn. Sys. B, 17(2012), 1585–1603.Google Scholar
[33]Zhao, W., Wang, J., and Peng, S., Error estimates of the θ-scheme for backward Stochastic differential equations, Dis. Con. Dyn. Sys. B, 12(2009), 905–924.Google Scholar
[34]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), 1563–1581.Google Scholar
[35]Zhao, W., Zhang, G., Ju, L., A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48(2010), 1369–1394.Google Scholar