Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs. (November 2015)

• Record Type: Journal Article
• Authors: Kong, Tao; Zhao, Weidong; Zhou, Tao
• Is Part Of: Communications in computational physics. Volume 18:Number 5(2015:Nov.)
• Page Start: 1482
• Page End: 1503
• Publication Date: 2015-11
• Subjects: 60H35, -- 65H20, -- 65H30; Fully nonlinear parabolic PDEs, -- second order FBSDEs, -- probabilistic interpretations, -- probabilistic numerical schemes; Mathematical physics -- Data processing -- Periodicals; Physics -- Data processing -- Periodicals; 530.150285
• DOI: 10.4208/cicp.240515.280815a
• Languages: English
• ISSNs: 1815-2406

Abstract

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.