Skip to main content
×
Home
    • Aa
    • Aa

High Order Numerical Schemes for Second-Order FBSDEs with Applications to Stochastic Optimal Control

  • Weidong Zhao (a1), Tao Zhou (a2) and Tao Kong (a1)
Abstract
Abstract

This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve the second-order FBSDEs (2FBSDEs). The key feature of the multistep schemes is that the Euler method is used to discretize the forward SDE, which dramatically reduces the entire computational complexity. Moreover, it is shown that the usual quantities of interest (e.g., the solution tuple (Yt ,Zt ,At t ) of the 2FBSDEs) are still of high order accuracy. Several numerical examples are given to show the effectiveness of the proposed numerical schemes. Applications of our numerical schemes to stochastic optimal control problems are also presented.

Copyright
Corresponding author
*Corresponding author. Email addresses: tzhou@lsec.cc.ac.cn (T. Zhou), wdzhao@sdu.edu.cn (W. Zhao)
References
Hide All
[1] BenderC. and ZhangJ., Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab., 18(2008), pp. 143177.
[2] BismutJ.M., Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl., 44(1973), pp. 384404.
[3] BouchardB. and TouziN., Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 111(2004), pp. 175206.
[4] ChassagneuxJ.F. and CrisenD., Runge-Kutta schemes for BSDEs, Ann. Appl. Probab., 24(2), 2014, pp. 679720.
[5] CheriditoP., SonerH. M., TouziN., and VictoirNicolas, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Communications on Pure and Applied Mathematics, Vol. LX(2007), pp. 10811110.
[6] DouglasJ., MaJ. and ProtterP., Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6(1996), pp. 940968.
[7] FahimA., TouziN., and WarinX., A probabilistic numerical method for fully nonlinear parabolic PDEs, Ann. Appl. Probab., 4(2011), pp. 13221364.
[8] FuY., ZhaoW. and ZhouT., Multistep schemes for forward backward stochastic differential equations with jumps, J. Sci. Comput., 69(2016), pp. 651672.
[9] FuY., ZhaoW., and ZhouT., Efficient sparse grid approximations for multi-dimensional coupled forward backward stochastic differential equations, submitted, 2015.
[10] GuoW., ZhangJ., and ZhuoJ., A Monotone Scheme for High Dimensional Fully Nonlinear PDEs, Ann. Appl. Probab., 25(2015), 15401580.
[11] EL KarouiN., PengS., and QuenezM. C., Backward stochastic differential equations in finance, Math. Finance, 7(1997), pp. 171.
[12] MaJ., ProtterP., and YongJ., Solving forward-backward stochastic differential equations explicitly – a four step scheme, Probab. Theory Related Fields, 98(1994), pp. 339359.
[13] MaJ., ShenJ., and ZhaoY., On numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal., 46(2008), pp. 26362661.
[14] MaJ. and YongJ., Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics, vol. 1702. Berlin: Springer.
[15] MilsteinG. N. and TretyakovM. V., Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28(2006), pp. 561582.
[16] OksendalB., Stochastic Differential Equations, Six Edition, Springer-Verlag, Berlin, 2003.
[17] PardouxE. and PengS., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14(1990), pp. 5561.
[18] PengS., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Repts., 37(1991), pp. 6174.
[19] SonerH.M., TouziN., and ZhangJ., Wellposedness of second order backward SDEs, Probab. Theory Relat. Fields, Vol. 153(2012), pp. 149190.
[20] TangT., ZhaoW., and ZhouT., Highly accurate numerical schemes for forward backward stochastic differential equations based on deferred correction approach, submitted, 2015.
[21] ZhangJ., A numerical scheme for BSDEs, Ann. Appl. Probab., 14(2004), pp. 459488.
[22] ZhaoW., ChenL. and PengS., A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput., 28(2006), pp. 15631581.
[23] ZhaoW., FuY., and ZhouT., New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput., 36(4), 2014, pp. A17311751.
[24] ZhaoW., ZhangG. and JuL., A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48(2010), pp. 13691394.
[25] ZhaoW., ZhangW. and JuL., A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations, Commun. Comput. Phys., 15(2014), pp. 618646.
[26] ZhaoW., ZhangW. and JuL., A multistep scheme for decoupled forward-backward stochastic differential equations, Numer. Math. Theory Methods Appl., 9(2), 2016, pp. 262288.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 27 *
Loading metrics...

Abstract views

Total abstract views: 103 *
Loading metrics...

* Views captured on Cambridge Core between 7th February 2017 - 21st October 2017. This data will be updated every 24 hours.