Choice of $\theta$ and mean-square exponential stability in the stochastic theta method of stochastic differential equations

Abstract

This paper examines the relationship of choice of $\theta$ and mean-square exponential stability in the stochastic theta method (STM) of stochastic differential equations (SDEs) and mainly includes the following three results: (i) under the linear growth condition for the drift term, when $\theta \in [0,1/2)$, the STM may preserve the mean-square exponential stability of the exact solution, but the counterexample shows that the STM cannot reproduce this stability without this linear growth condition; (ii) when $\theta \in (1/2,1)$, without the linear growth condition for the drift term, the STM may reproduce the mean-square exponential stability of the exact solution, but the bound of the Lyapunov exponent cannot be preserved; (iii) when $\theta =1$ (this STM is called as the backward Euler–Maruyama (BEM) method), the STM can reproduce not only the mean-square exponential stability, but also the bound of the Lyapunov exponent. This paper also gives the sufficient and necessary conditions of the mean-square exponential stability of the STM for the linear SDE when $\theta \in [0,1/2)$ and $\theta \in [1/2,1]$, respectively, and the simulations also illustrate these theoretical results.