随机微分方程稳定性的新的充分条件

The new sufficient condition of stability for stochastic differential equation

考虑随机微分方程(SDEs)相容解的几种随机稳定性。Gard和Mao分别应用Lya-punov第二方法给出了保证It型随机微分方程(SDEs)的相容解是随机稳定、随机渐近稳定及全局随机渐近稳定的充分条件,这些条件通常要求Lyapunov函数V(x,t)为正定函数。应用随机分析的技巧,在很宽的条件下,把Lyapunov函数V(x,t)正定的条件去掉,且仍然保证方程的解的几种随机稳定性。结果推广了随机微分方程稳定性的经典结果。

Consider three types of stochastic stabilities for adapted solutions of stochastic differential equations＇ （SDEs）, namely, stochastically stability, stochastically asymptotically stability and globally stochastically asymp- totically stability. By using Lyapunov＇ s second method, Gard and Mao have given sufficient conditions for these three stabilities. However, their Lyapunov functions are supposed to be positive definite. Some loose conditions （ i. e., Lyapunov functions are not necessary to be positive definite） for Lyapunov functions are given by applying sto- chastic analysis technique. This is strict generalizations of the classical results.