Convergence and qualitative properties of modified explicit schemes for BSDEs with polynomial growth

TL;DR

This paper develops a framework for analyzing modified explicit schemes for BSDEs with polynomial growth, enabling convergence and qualitative property preservation with reduced computational effort compared to implicit schemes.

Contribution

It introduces a systematic framework for analyzing modified explicit schemes for BSDEs with polynomial growth, achieving convergence and qualitative properties similar to implicit schemes.

Findings

Modified explicit schemes can converge with the same rate as implicit schemes.

The framework ensures preservation of qualitative properties like comparison theorems.

Modified explicit schemes can outperform implicit schemes computationally.

Abstract

The theory of Forward-Backward Stochastic Differential Equations (FBSDEs) paves a way to probabilistic numerical methods for nonlinear parabolic PDEs. The majority of the results on the numerical methods for FBSDEs relies on the global Lipschitz assumption, which is not satisfied for a number of important cases such as the Fisher--KPP or the FitzHugh--Nagumo equations. Furthermore, it has been shown in \cite{LionnetReisSzpruch2015} that for BSDEs with monotone drivers having polynomial growth in the primary variable $y$, only the (sufficiently) implicit schemes converge. But these require an additional computational effort compared to explicit schemes. This article develops a general framework that allows the analysis, in a systematic fashion, of the integrability properties, convergence and qualitative properties (e.g.~comparison theorem) for whole families of modified explicit schemes. The framework yields the convergence of some modified explicit scheme with the same rate as implicit schemes and with the computational cost of the standard explicit scheme. To illustrate our theory, we present several classes of easily implementable modified explicit schemes that can computationally outperform the implicit one and preserve the qualitative properties of the solution to the BSDE. These classes fit into our developed framework and are tested in computational experiments.