Mean Square Polynomial Stability of Numerical Solutions to a Class of Stochastic Differential Equations

TL;DR

This paper investigates how well the Euler--Maruyama and backward Euler--Maruyama methods reproduce the polynomial decay stability of certain stochastic differential equations, using gamma function estimates.

Contribution

It provides the first analysis of polynomial stability preservation for these numerical methods applied to SDEs.

Findings

Euler--Maruyama method reproduces polynomial decay stability

Backward Euler--Maruyama method also preserves polynomial stability

Analysis relies on gamma function estimates

Abstract

The exponential stability of numerical methods to stochastic differential equations (SDEs) has been widely studied. In contrast, there are relatively few works on polynomial stability of numerical methods. In this letter, we address the question of reproducing the polynomial decay of a class of SDEs using the Euler--Maruyama method and the backward Euler--Maruyama method. The key technical contribution is based on various estimates involving the gamma function.