一类标量线性随机时滞微分方程Euler-Maruyama方法的均方稳定性判据

Mean-Square Stability Criteria for the Euler-Maruyama Method Applied to a Class of Scalar Linear Stochastic Delay Differential Equations

摘要: 本文在已有随机时滞微分方程数值稳定性研究的基础上，研究一类标量线性随机时滞微分方程Euler-Maruyama方法的均方稳定性，重点给出形式简单、可直接检验的显式稳定性充分判据，并分析数值方法在充分小步长下对解析均方指数稳定性的保持性质。首先，利用Itô公式和Halanay型不等式，给出原方程零解均方指数稳定的一个充分条件。其次，在时滞长度为步长整数倍的情形下，构造Euler-Maruyama数值格式，并通过二阶矩递推估计得到数值解均方指数稳定的显式步长条件。进一步证明，当原方程满足该解析稳定性条件时，Euler-Maruyama方法在充分小且与时滞相容的步长下能够保持均方指数稳定性。最后，通过Monte Carlo数值实验说明步长和噪声强度对数值稳定性的影响。数值结果表明，本文所得判据虽为充分条件，但能够为随机时滞微分方程数值方法的稳定性保持分析提供保守且可操作的检验准则。

Abstract: Building on existing studies of numerical stability for stochastic delay differential equations, this paper studies the mean-square stability of the Euler-Maruyama method for a scalar linear stochastic delay differential equation. The aim is to present simple, explicit and directly verifiable sufficient stability criteria and to analyze the stability-preserving property of the numerical method under sufficiently small stepsizes. By using Itô’s formula and a Halanay-type inequality, a sufficient condition for the mean-square exponential stability of the zero solution is first obtained. Under the assumption that the delay length is an integer multiple of the stepsize, the Euler-Maruyama scheme is constructed, and a second-moment recurrence estimate yields an explicit stepsize condition for mean-square exponential stability of the numerical solution. It is further proved that, whenever the exact equation satisfies the proposed stability condition, the Euler-Maruyama method preserves this stability for all sufficiently small admissible stepsizes. Monte Carlo experiments illustrate the effects of the stepsize and the noise intensity on the numerical mean-square behavior. The results show that the proposed criteria, although sufficient rather than necessary, provide a conservative and practical framework for stability-preserving analysis of numerical methods for stochastic delay differential equations.