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Volume 18, Issue 5
Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs

Tao Kong, Weidong Zhao & Tao Zhou

DOI: 10.4208/cicp.240515.280815a

Commun. Comput. Phys., 18 (2015), pp. 1482-1503.

Published online: 2018-04

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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.

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@Article{CiCP-18-1482, author = {Tao Kong, Weidong Zhao and Tao Zhou}, title = {Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs}, journal = {Communications in Computational Physics}, year = {2018}, volume = {18}, number = {5}, pages = {1482--1503}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.240515.280815a}, url = {http://global-sci.org/intro/article_detail/cicp/11077.html} }
TY - JOUR T1 - Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs AU - Tao Kong, Weidong Zhao & Tao Zhou JO - Communications in Computational Physics VL - 5 SP - 1482 EP - 1503 PY - 2018 DA - 2018/04 SN - 18 DO - http://doi.org/10.4208/cicp.240515.280815a UR - https://global-sci.org/intro/article_detail/cicp/11077.html KW - AB -

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.

Tao Kong, Weidong Zhao and Tao Zhou. (2018). Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs. Communications in Computational Physics. 18 (5). 1482-1503. doi:10.4208/cicp.240515.280815a
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