A monotone scheme for high-dimensional fully nonlinear PDEs

TL;DR

This paper introduces a feasible monotone numerical scheme for solving high-dimensional fully nonlinear parabolic PDEs, extending previous methods by relaxing key constraints and demonstrating effectiveness up to 12 dimensions.

Contribution

It proposes a new monotone scheme that relaxes previous constraints, enabling efficient solutions for high-dimensional fully nonlinear PDEs, including special cases like coupled FBSDEs.

Findings

Effective up to 12 dimensions in numerical tests

Weakens constraints on generator dependence on Hessian

Builds on and extends the monotone scheme paradigm

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.