一些稳定的随机系统是非指数稳定的，可能出现方程解收敛的速率较指数速率低的情况，如多项式或对数．为了更精确地定量分析系统的稳定性，文中将指数稳定概念推广到更一般稳定的衰减函数，研究了一类马尔可夫调制随机系统在一般衰减速率下的稳定性．利用ItÔ公式、Borel-Cantelli 引理和鞅指数不等式等随机分析技巧，先建立了解析解p 阶矩φ( t) 稳定和几乎必然φ( t) 稳定的定理，然后证明了在相同的条件下，对足够小的步长Δ，Euler Maruyama 方法能保持相同的稳定性．

It is worth pointing out that some stochastic systems are indeed stable but subject to a certain lower decay rate which is different from exponential decay,such as polynomial or logarithmic. For more accurate quantitative analyses of stability properties,this paper extends the usual exponential stability concepts to a more general stable decay function and investigates the general decay stability of stochastic differential equations with Markovian switching. Firstly,some φ( t) -stability criteria in p-th moment and almost surely sense for the analytical solutions are established,by utilizing ItÔ formula,Borel-Cantelli and martingale exponential inequalities. Then the Euler Maruyama method is shown to be effective in capturing φ( t) -stability behavior for all sufficiently small timesteps under appropriate conditions.