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Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs

Part of: Nonlinear algebraic or transcendental equations Stochastic analysis

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)
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Abstract

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.

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

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