非线性随机延迟微分方程半隐式Euler方法的均方稳定性

MEAN-SQUARE STABILITY OF SEMI-IMPLICIT EULER 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-稳定推广到一般情形,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,半隐式Euler方法是MS-稳定的且带线性插值的半隐式Euler方法是GMS-稳定的理论结果.

In this paper, the authors investigated the mean-square stability of semi-implicit Euler 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 zero solution satisfies the sufficient condition of the mean square stability, we obtained several theoretical ruslts of semi-implicit Euler methods. If the drift term satisfies some restrictions, then semi-implicit Euler methods is MS-stable, moreover, semi-implicit Euler methods with linear interpolation procedure is GMS-stable.