Article2015

Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs

25 citations0/0 files storedCommunications in Computational Physics

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Stochastic processes and financial applications Numerical methods for differential equations Fuzzy Systems and Optimization

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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References

1987Stochastic Differential Equations: An Introduction with Applications.2006Stochastic Differential Equations—An Introduction With Applications 2005Backward stochastic differential equations and quasilinear parabolic partial differential equations 2019Nonlinear Expectations and Stochastic Calculus under Uncertainty 2004Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations 1991Probabilistic interpretation for systems of quasilinear parabolic partial differential equations 2005A regression-based Monte Carlo method to solve backward stochastic differential equations 1999Forward-backward stochastic differential equations and quasilinear parabolic PDEs

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