Adapted $\theta$-Scheme and Its Error Estimates for Backward Stochastic Differential Equations

TL;DR

This paper introduces a novel high-order numerical scheme for backward stochastic differential equations that adaptively adjusts parameters to reduce errors, verified through theoretical error estimates and numerical experiments.

Contribution

It proposes an adaptive $ heta$-scheme for BSDEs that improves accuracy by reducing truncation errors based on integrand characteristics.

Findings

Error estimates confirm the scheme's high order accuracy

Numerical experiments verify the theoretical error bounds

Adaptive $ heta$-scheme outperforms traditional methods

Abstract

In this paper we propose a new kind of high order numerical scheme for backward stochastic differential equations(BSDEs). Unlike the traditional $\theta$-scheme, we reduce truncation errors by taking $\theta$ carefully for every subinterval according to the characteristics of integrands. We give error estimates of this nonlinear scheme and verify the order of scheme through a typical numerical experiment.