Equivalence of pth moment stability between stochastic differential delay equations and their numerical methods

TL;DR

This paper establishes a theoretical equivalence between the stability of stochastic differential delay equations and their numerical methods in the pth moment sense, under certain convergence and boundedness conditions.

Contribution

It proves a general theorem linking the pth moment stability of SDDEs and their numerical solutions, expanding applicability to the truncated Euler-Maruyama method.

Findings

Theorem demonstrating stability equivalence under broad conditions

Application of the theorem to the truncated Euler-Maruyama method

Relaxed step size requirements compared to previous work

Abstract

In this paper, a general theorem on the equivalence of pth moment stability between stochastic differential delay equations (SDDEs) and their numerical methods is proved under the assumptions that the numerical methods are strongly convergent and have the bouneded $p$th moment in the finite time. The truncated Euler-Maruyama (EM) method is studied as an example to illustrate that the theorem indeed covers a large ranges of SDDEs. Alongside the investigation of the truncated EM method, the requirements on the step size of the method are significantly released compared with the work, where the method was initially proposed.