Stability Analysis for the Numerical Simulation of Hybrid Stochastic Differential Equations (Journal of Nonlinear Modeling and Analysis, 2024)
This paper is mainly concerned with the exponential stability ofa class of hybrid stochastic differential equations–stochastic differential equations with Markovian switching (SDEwMSs). It first devotes to revealing thatunder 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 suitablestep 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 sizesimplies the equation is almost surely exponentially stable or the correspondingIEM method, respectively. In that sense, one can infer that the SDEwMS isalmost surely exponentially stable and the IEM method, no matter whetherthe SDEwMS is  $p$th moment exponentially stable or the IEM method. Anexample is demonstrated to illustrate the obtained results.