Hostname: page-component-76c49bb84f-ndtt8 Total loading time: 0 Render date: 2025-07-10T22:50:01.827Z Has data issue: false hasContentIssue false
  • English
  • Français

Probabilistic High Order Numerical Schemes for Fully NonlinearParabolic PDEs

Part of: Stochastic analysis Nonlinear algebraic or transcendental equations

Published online by Cambridge University Press:  23 November 2015

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

Abstract

In this paper, we are concerned with probabilistic high order numerical schemesfor Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs,it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exactsolutions admit probabilistic interpretations, i.e., the solution of a fullynonlinear parabolic PDE solves a corresponding second order forward backwardstochastic differential equation (2FBSDEs). Our numerical schemes rely onsolving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T.Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in ournumerical schemes, one has the flexibility to choose the associated forward SDE,and a suitable choice can significantly reduce the computational complexity.Various numerical examples including the HJB equations are presented to show theeffectiveness and accuracy of the proposed numerical schemes.

Keywords

Fully nonlinear parabolic PDEssecond order FBSDEsprobabilistic interpretationsprobabilistic numerical schemes

MSC classification

Secondary: 60H35: Computational methods for stochastic equations 65H20: Global methods, including homotopy approaches

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1482 - 1503
Copyright
Copyright © Global-Science Press 2015 

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.)

Article purchase

Temporarily unavailable

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]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
[3]Chassagneux, J.F. and Crisen, D., Runge-Kutta schemes for BSDEs, to appear in Ann. Appl. Probab., 2014.Google Scholar
[4]Cheridito, P., Soner, H. M., Touuzi, 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.CrossRefGoogle Scholar
[5]Crisan, D. and Manolarakis, K., Solving backward stochastic differential equations using the cubature method, SIAM J. Math. Finance, (3)2012, pp. 534–571.Google Scholar
[6]Delarue, F. and Menozzi, S., A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab., 16(2006), pp. 140–184.CrossRefGoogle Scholar
[7]Delarue, F., and Menozzi, S., An interpolated stochastic algorithm for quasi-linear pdes. Mathematics of Computation 77, 261 (2008), 125–158.Google Scholar
[8]Douglas, J.,Ma, J. and Protter, P., Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6(1996), pp. 940–968.Google Scholar
[9]Fahim, Arash, Touzi, Nizar,and Warin, Xavier, A probabilistic numerical method for fully nonlinear parabolic PDEs, Ann. Appl. Probab., 4(2011), pp. 1322–1364.Google Scholar
[10]Feng, X., Glowinski, R., and Neilan, M., Recent developments in numerical methods for fully nonlinear second order partial differential equations. SIAM Review 55, 2(2013), 205–267.Google Scholar
[11]Fu, Y., Zhao, W., and Zhou, T., Efficient sparse grid approximations for multi-dimensional coupled forward backward stochastic differential equations, submitted, 2015.Google Scholar
[12]Guo, W.,Zhang, J., and Zhuo, J., A Monotone Scheme for High Dimensional Fully Nonlinear PDEs, arXiv:1212.0466, to appear in Ann. Appl. Probab., 2015.Google Scholar
[13]Kong, Tao, Zhao, Weidong, and Zhou, Tao, High order numerical schemes for second order FBSDEs with applications to stochastic optimal control, arXiv:1502.03206,2015.Google Scholar
[14]øksendal, Bernt, Stochastic Differential Equations: An Introduction with Applications, 6th edition (2014) Springer.Google Scholar
[15]Lemor, J. P., Gobet, E. and Warin, X., A regression-based Monte Carlo method for backward stochastic differential equations, Ann. Appl. Probab., 15(2005), pp. 2172–2202.Google Scholar
[16]Milstein, N. G. and Tretyakov, M. V., Discretization of Forward-Backward Stochastic Differential Equations And Related Quasi-linear Parabolic Equations, SIAM J. Numer. Anal, 27(2007), 24–34.Google Scholar
[17]Pardoux, E. and Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in CIS, Springer, 176 (1992), 200–217.Google Scholar
[18]Pardoux, E. and Tang, S., Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Relat. Fields, 114 (1999), pp. 123–150.Google Scholar
[19]Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, preprint (2010), arXiv:1002.4546v1.Google Scholar
[20]Peng, S. G., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Repts., 37 (1991), pp. 61–74.Google Scholar
[21]Soner, H. M., Touzi, N., and Zhang, J., Wellposedness of second order backward SDEs, Probab. Theory Relat. Fields, Vol. 153 (2012), pp:149–190.Google Scholar
[22]Tan, X., Probabilistic Numerical Approximation for Stochastic Control Problems, preprint, 2011.Google Scholar
[23]Tan, X., A splitting method for fully nonlinear degenerate parabolic PDEs, preprint, 2011.Google Scholar
[24]Tang, T., Zhao, W., and Zhou, T., Deferred correction methods for forward backward stochastic differential equations, submitted, 2015.Google Scholar
[25]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
[26]W.Zhao, , 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), pp. A1731–1751, 2014.Google Scholar
[27]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
[28]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.Google Scholar