非线性随机延迟微分方程Euler-Maruyama方法的均方稳定性

MEAN-SQUARE STABILITY OF EULER-MARUYAMA METHODS FOR NONLINEAR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

本文首先将数值方法的均方稳定性的概念MS-稳定与GMS-稳定从线性试验方程推广到一般非线性的情形,然后针对一维情形下的非线性随机延迟微分方程初值问题,证明了如果问题本身满足零解是均方渐近稳定的充分条件,那么当漂移项满足一定的限制条件时,Euler- Maruyama方法是MS-稳定的与带线性插值的Euler-Maruyama方法是GMS-稳定的理论结果．

In this paper, the authors investigated the mean-square stability of Euler- Maruyama methods for the nonlinear stochastic delay differential equations. At first, the both definitions of MS-stability and GMS-stability of numerical methods are developed from the linear scalar system to general case. And then, when the analytical solution satisfies the sufficient condition of the mean square stability, we obtained several theoretical results of Euler-Maruyama methods. If the drift term satisfies some restrictions, then Euler-Maruyama methods is MS-stable and Euler-Maruyama methods with linear interpolation is GMS-stable.