A monotone scheme for high-dimensional fully nonlinear PDEs

Abstract

In this paper we propose a feasible numerical scheme for high-dimensional, fully nonlinear parabolic PDEs, which includes the quasi-linear PDE associated with a coupled FBSDE as a special case. Our paper is strongly motivated by the remarkable work Fahim, Touzi and Warin [Ann. Appl. Probab. 21 (2011) 1322-1364] and stays in the paradigm of monotone schemes initiated by Barles and Souganidis [Asymptot. Anal. 4 (1991) 271-283]. Our scheme weakens a critical constraint imposed by Fahim, Touzi and Warin (2011), especially when the generator of the PDE depends only on the diagonal terms of the Hessian matrix. Several numerical examples, up to dimension 12, are reported.

Published at http://dx.doi.org/10.1214/14-AAP1030 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• A regression-based Monte Carlo method to solve backward stochastic differential equations
• Nonlinear Expectations and Stochastic Calculus under Uncertainty
• Time discretization and Markovian iteration for coupled FBSDEs
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