Skip to main content Accesibility Help

Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs

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

In this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving 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 our numerical 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 the effectiveness and accuracy of the proposed numerical schemes.

Corresponding author
*Corresponding author. Email addresses: (T. Kong), (W. Zhao), (T. Zhou)
Hide All
[1]Bender, C. and Zhang, J., Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab., 18(2008), pp. 143177.
[2]Bouchard, B. and Touzi, N., Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stochastic Process. Appl., 111(2004), pp.175206.
[3]Chassagneux, J.F. and Crisen, D., Runge-Kutta schemes for BSDEs, to appear in Ann. Appl. Probab., 2014.
[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. 10811110.
[5]Crisan, D. and Manolarakis, K., Solving backward stochastic differential equations using the cubature method, SIAM J. Math. Finance, (3)2012, pp. 534571.
[6]Delarue, F. and Menozzi, S., A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab., 16(2006), pp. 140184.
[7]Delarue, F., and Menozzi, S., An interpolated stochastic algorithm for quasi-linear pdes. Mathematics of Computation 77, 261 (2008), 125158.
[8]Douglas, J.,Ma, J. and Protter, P., Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6(1996), pp. 940968.
[9]Fahim, Arash, Touzi, Nizar,and Warin, Xavier, A probabilistic numerical method for fully nonlinear parabolic PDEs, Ann. Appl. Probab., 4(2011), pp. 13221364.
[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), 205267.
[11]Fu, Y., Zhao, W., and Zhou, T., Efficient sparse grid approximations for multi-dimensional coupled forward backward stochastic differential equations, submitted, 2015.
[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.
[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.
[14]øksendal, Bernt, Stochastic Differential Equations: An Introduction with Applications, 6th edition (2014) Springer.
[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. 21722202.
[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), 2434.
[17]Pardoux, E. and Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in CIS, Springer, 176 (1992), 200217.
[18]Pardoux, E. and Tang, S., Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Relat. Fields, 114 (1999), pp. 123150.
[19]Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, preprint (2010), arXiv:1002.4546v1.
[20]Peng, S. G., Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Repts., 37 (1991), pp. 6174.
[21]Soner, H. M., Touzi, N., and Zhang, J., Wellposedness of second order backward SDEs, Probab. Theory Relat. Fields, Vol. 153 (2012), pp:149190.
[22]Tan, X., Probabilistic Numerical Approximation for Stochastic Control Problems, preprint, 2011.
[23]Tan, X., A splitting method for fully nonlinear degenerate parabolic PDEs, preprint, 2011.
[24]Tang, T., Zhao, W., and Zhou, T., Deferred correction methods for forward backward stochastic differential equations, submitted, 2015.
[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. 15631581.
[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. A17311751, 2014.
[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. 13691394.
[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. 618646.
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? *


MSC classification


Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed