Stability Analysis for the Numerical Simulation of Hybrid Stochastic Differential Equations

Abstract

This paper is mainly concerned with the exponential stability of a class of hybrid stochastic differential equations–stochastic differential equations with Markovian switching (SDEwMSs). It first devotes to revealing that under the global Lipschitz condition, a SDEwMS is $p$th ($p ∈ (0, 1)$) moment exponentially stable if and only if its corresponding improved Euler-Maruyama (IEM) method is $p$th moment exponentially stable for a suitable step size. It then shows that the SDEwMS is $p$th ($p ∈ (0, 1)$) moment exponentially stable or its corresponding IEM method with small enough step sizes implies the equation is almost surely exponentially stable or the corresponding IEM method, respectively. In that sense, one can infer that the SDEwMS is almost surely exponentially stable and the IEM method, no matter whether the SDEwMS is  $p$th moment exponentially stable or the IEM method. An example is demonstrated to illustrate the obtained results.