一类线性随机时滞微分方程半隐式 Euler-Maruyama方法的显式均方指数 稳定性充分判据

Square Exponential Stability of a Semi-Implicit Euler-Maruyama Method for a Class of Linear Stochastic Delay Differential Equations

摘要: 本文研究一类标量线性随机时滞微分方程半隐式Euler-Maruyama方法的均方稳定性。针对含线性耗散、离散时滞反馈和乘性噪声的线性测试模型，先由Itô公式与Halanay型不等式给出连续随机时滞微分方程零解均方指数稳定的显式充分条件，再构造带参数$\theta \in \left[0,1\right]$ 的半隐式Euler-Maruyama格式，其中当前漂移项按θ-隐式方式处理，时滞项和扩散项保持显式。通过二阶矩递推估计，得到数值均方指数稳定的可检验充分判据q_θ(h) < 1。进一步证明，当原方程满足解析稳定性条件时，该格式在充分小且与时滞相容的步长下保持均方指数稳定性。需要说明的是，本文判据属于保守的二阶矩上界型充分条件，旨在提供一个便于直接计算和数值验证的稳定性分析框架，并不声称给出最优稳定域或充要条件。数值实验在所选参数下展示了半隐式参数对由本文充分判据刻画的可保证稳定区域的改善作用。

Abstract: This paper investigates the mean-square stability of a semi-implicit Euler-Maruyama method for a scalar linear stochastic delay differential equation with linear dissipation, discrete delay feedback and multiplicative noise. An explicit sufficient condition for the mean-square exponential stability of the zero solution of the underlying equation is first obtained by Itô’s formula and a Halanay-type inequality. A semi-implicit Euler-Maruyama scheme with parameter $\theta \in \left[0,1\right]$ is then constructed, where the current drift term is treated in a θ-implicit manner while the delay and diffusion terms are treated explicitly. A directly verifiable sufficient criterion q_θ(h) < 1 for mean-square exponential stability is derived from a second-moment recurrence estimate. Further proof demonstrates that when the original equation satisfies the analytic stability condition, the scheme maintains mean-square exponential stability with a sufficiently small step size compatible with time delay. It should be noted that the criterion presented in this paper is a conservative second-moment upper bound sufficient condition, aiming to provide a stability analysis framework that facilitates direct calculation and numerical verification, without claiming to give an optimal stability region or necessary and sufficient conditions. Numerical experiments, under the selected parameters, demonstrate the improvement effect of semi-implicit parameters on the guaranteed stability region characterized by the sufficient criterion presented in this paper.