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

Part of: Stochastic analysis Nonlinear algebraic or transcendental equations

Published online by Cambridge University Press:  07 February 2017

Weidong Zhao ,
Tao Zhou  and
Tao Kong
Weidong Zhao
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China
Tao Zhou*
Affiliation:
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Tao Kong*
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China
*
*Corresponding author. Email addresses:tzhou@lsec.cc.ac.cn (T. Zhou), wdzhao@sdu.edu.cn (W. Zhao)
*Corresponding author. Email addresses:tzhou@lsec.cc.ac.cn (T. Zhou), wdzhao@sdu.edu.cn (W. Zhao)
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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,Γt) 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.

Keywords

Multistep schemessecond-order forward backward stochastic differential equationsstochastic optimal control

MSC classification

Secondary: 60H35: Computational methods for stochastic equations 65H20: Global methods, including homotopy approaches
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 3 , March 2017 , pp. 808 - 834
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Bender, C. and Zhang, J., Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab., 18(2008), pp. 143–177.CrossRefGoogle Scholar
[2] Bismut, J.M., Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl., 44(1973), pp. 384–404.CrossRefGoogle 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.Google Scholar
[4] Chassagneux, J.F. and Crisen, D., Runge-Kutta schemes for BSDEs, Ann. Appl. Probab., 24(2), 2014, pp. 679–720.Google Scholar
[5] Cheridito, P., Soner, H. M., Touzi, N., and Victoir, Nicolas, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Communications on Pure and Applied Mathematics, Vol. LX(2007), pp. 1081–1110.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.CrossRefGoogle Scholar
[7] Fahim, A., Touzi, N., and Warin, X., A probabilistic numerical method for fully nonlinear parabolic PDEs, Ann. Appl. Probab., 4(2011), pp. 1322–1364.Google Scholar
[8] Fu, Y., Zhao, W. and Zhou, T., Multistep schemes for forward backward stochastic differential equations with jumps, J. Sci. Comput., 69(2016), pp. 651–672.Google Scholar
[9] Fu, Y., Zhao, W., and Zhou, T., Efficient sparse grid approximations for multi-dimensional coupled forward backward stochastic differential equations, submitted, 2015.Google Scholar
[10] Guo, W., Zhang, J., and Zhuo, J., A Monotone Scheme for High Dimensional Fully Nonlinear PDEs, Ann. Appl. Probab., 25(2015), 1540–1580.CrossRefGoogle Scholar
[11] EL Karoui, N., Peng, S., and Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 7(1997), pp. 1–71.Google Scholar
[12] Ma, J., Protter, P., and Yong, J., Solving forward-backward stochastic differential equations explicitly – a four step scheme, Probab. Theory Related Fields, 98(1994), pp. 339–359.CrossRefGoogle Scholar
[13] Ma, J., Shen, J., and Zhao, Y., On numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal., 46(2008), pp. 2636–2661.CrossRefGoogle Scholar
[14] Ma, J. and Yong, J., Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Mathematics, vol. 1702. Berlin: Springer.Google Scholar
[15] Milstein, G. N. and Tretyakov, M. V., Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28(2006), pp. 561–582.CrossRefGoogle Scholar
[16] Oksendal, B., Stochastic Differential Equations, Six Edition, Springer-Verlag, Berlin, 2003.Google Scholar
[17] Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14(1990), pp. 55–61.Google Scholar
[18] Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Repts., 37(1991), pp. 61–74.Google Scholar
[19] Soner, H.M., Touzi, N., and Zhang, J., Wellposedness of second order backward SDEs, Probab. Theory Relat. Fields, Vol. 153(2012), pp. 149–190.CrossRefGoogle Scholar
[20] Tang, T., Zhao, W., and Zhou, T., Highly accurate numerical schemes for forward backward stochastic differential equations based on deferred correction approach, submitted, 2015.Google Scholar
[21] Zhang, J., A numerical scheme for BSDEs, Ann. Appl. Probab., 14(2004), pp. 459–488.Google Scholar
[22] 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
[23] 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.CrossRefGoogle Scholar
[24] 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.Google Scholar
[25] Zhao, W., Zhang, W. and Ju, L., A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations, Commun. Comput. Phys., 15(2014), pp. 618–646.CrossRefGoogle Scholar
[26] Zhao, W., Zhang, W. and Ju, L., A multistep scheme for decoupled forward-backward stochastic differential equations, Numer. Math. Theory Methods Appl., 9(2), 2016, pp. 262–288.CrossRefGoogle Scholar