Articlebronze2015

A monotone scheme for high-dimensional fully nonlinear PDEs

53 citations0/0 files storedThe Annals of Applied Probability

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Stochastic processes and financial applications Advanced Numerical Methods in Computational Mathematics Nonlinear Partial Differential Equations

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.

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References

2001Valuing American Options by Simulation: A Simple Least-Squares Approach 2005Backward stochastic differential equations and quasilinear parabolic partial differential equations 1994Solving forward-backward stochastic differential equations explicitly — a four step scheme 2019Nonlinear Expectations and Stochastic Calculus under Uncertainty 2004Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations 2005A regression-based Monte Carlo method to solve backward stochastic differential equations 2004A numerical scheme for BSDEs 2002An analysis of a least squares regression method for American option pricing

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