Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs (Communications in Computational Physics, 2020)
In this paper, we are concerned with probabilistic high order numericalschemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolicPDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exactsolutions admit probabilistic interpretations, i.e., the solution of a fully nonlinearparabolic PDE solves a corresponding second order forward backward stochastic differentialequation (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 flexibilityto choose the associated forward SDE, and a suitable choice can significantlyreduce the computational complexity. Various numerical examples including the HJBequations are presented to show the effectiveness and accuracy of the proposed numericalschemes.