An Efficient Numerical Method for Forward-Backward Stochastic Differential Equations Driven by $G$-Brownian motion

TL;DR

This paper introduces an efficient numerical method for solving forward-backward stochastic differential equations driven by $G$-Brownian motion, which are linked to fully nonlinear PDEs, with proven convergence and demonstrated accuracy.

Contribution

The paper develops new numerical schemes for $G$-FBSDEs, including an approximate conditional $G$-expectation, and provides rigorous error analysis and convergence proofs.

Findings

Numerical schemes accurately solve $G$-FBSDEs.

Convergence of the proposed methods is rigorously established.

Numerical experiments confirm the effectiveness of the methods.

Abstract

In this paper, we study the numerical method for solving forward-backward stochastic differential equations driven by $G$-Brownian motion ($G$-FBSDEs) which correspond to fully nonlinear partial differential equations (PDEs). First, we give an approximate conditional $G$-expectation and obtain feasible methods to calculate the distribution of $G$-Brownian motion. On this basis, some efficient numerical schemes for $G$-FBSDEs are then proposed. We rigorously analyze errors of the proposed schemes and prove the convergence results. Finally, several numerical experiments are given to demonstrate the accuracy of our method.