Mean-square convergence and stability of the backward Euler method for stochastic differential delay equations with highly nonlinear growing coefficients

TL;DR

This paper analyzes the mean-square convergence and stability of the backward Euler method for stochastic differential delay equations with nonlinear coefficients, providing error bounds, convergence rate, and stability inheritance proofs.

Contribution

It introduces a novel technique to establish convergence and stability of BEM for SDDEs with polynomially growing coefficients, without requiring bounded moments.

Findings

Mean-square convergence rate of 1/2 established

Error bounds for the backward Euler method derived

BEM inherits exponential mean-square stability under general conditions

Abstract

Over the last few decades, the numerical methods for stochastic differential delay equations (SDDEs) have been investigated and developed by many scholars. Nevertheless, there is still little work to be completed. By virtue of the novel technique, this paper focuses on the mean-square convergence and stability of the backward Euler method (BEM) for SDDEs whose drift and diffusion coefficients can both grow polynomially. The upper mean-square error bounds of BEM are obtained. Then the convergence rate, which is one-half, is revealed without using the moment boundedness of numerical solutions. Furthermore, under fairly general conditions, the novel technique is applied to prove that the BEM can inherit the exponential mean-square stability with a simple proof. At last, two numerical experiments are implemented to illustrate the reliability of the theories.