Stability Of Stochastic Differential Delay Equations With Markov Process

This paper mainly studies the stability of stochastic delay differential equations with markov process.It includes p th moment exponential stability,the almost sure exponential stability.In general,when studying the stability of such equations,we usually choose the appropriate Lyapunov function,but in reality,it is hard to find it.So in this case,we can try to find a numerical method.The stability of the original equation is discussed through the stability of this numerical method.Among many properties of numerical methods,stability has always been a concerned property.At the beginning of this paper,we briefly introduce the stochastic delay differential equation with markov process,and also study the stability of p th moment exponent of this equation in detail.By introducing a theta method,the continuous,if the coefficient of the equation can meet global Lipschitz condition,and can satisfy the linear-growth condition,then about p th moment exponential stability,we find the equation of the numerical solution and analytical solution of a link between.In other words,for p ?(0,1),when the step length is sufficiently small,the necessary and sufficient condition for the stability of the p th moment exponential is that the method is also p th moment exponential stable.Next,we study almost sure exponential stability of this equation.If the coefficient in the equation can satisfy some appropriate conditions,that we can show that its theta method is almost sure exponential stability.The results shows the research on the stability of this equation,if we cannot simply find a suitable Lyapunov function to research,so we can choose a kind of numerical methods to continue study.At the same time,we prove that for a sufficiently small p ?(0,1),if the theta method of this equation is p th moment exponential stability,then we can deduce that the original equation is also p th moment exponential stability.