Mean-square contractivity of stochastic $\theta$-methods

TL;DR

This paper analyzes the mean-square contractivity of stochastic -methods, establishing conditions under which numerical solutions preserve the exponential decay of differences in stochastic differential equations, with verified sharp step size restrictions.

Contribution

It provides a detailed stability analysis of stochastic -methods, deriving sharp step size bounds to ensure mean-square contractivity in numerical solutions.

Findings

Derived explicit step size restrictions for stability

Confirmed sharpness through numerical experiments

Demonstrated preservation of exponential decay in numerical solutions

Abstract

The paper is focused on the nonlinear stability analysis of stochastic $\theta$-methods. In particular, we consider nonlinear stochastic differential equations such that the mean-square deviation between two solutions exponentially decays, i.e., a mean-square contractive behaviour is visible along the stochastic dynamics. We aim to make the same property visible also along the numerical dynamics generated by stochastic $\theta$-methods: this issue is translated into sharp stepsize restrictions depending on parameters of the problem, here accurately estimated. A selection of numerical tests confirming the effectiveness of the analysis and its sharpness is also provided.