Markov Modulated Neutral Stochastic Differential Equations, Numerical Solution And The Distribution Of Stability

In this paper, we investigate the convergence of numerical solutions and stability in distribution of neutral stochastic differential delay equations with Markovian switching and Poisson jump.Explicit solutions can hardly be obtained for most of stochastic differential delay equations, except some linear stochastic differential delay equations. As a result ,several numerical schemes such as EulerMaruyama method ,backward Euler-Maruyama,θmethod, stochastic Taylor expansion e.t. have been developed to produce approximate solutions to study the stability of trivial solution and the convergence between numerical solution and exact solution.Secondly, most of literature are concerned with the stability of the trivial solution in moment or probability (i.e. the solution will tend to zero in moment or in probability).However, such stability is sometimes too strong while in many practical situations it is useful to know whether or not the probability distribution of the solution will converge to some probability distribution(but not necessary to zero).Such stability is called stability in distribution.The paper contains four chapters.Chapter 1 is the preface.In chapter 2, we investigate the convergence between numerical solutions and exact solutions of neutral stochastic differential delay equation with Markovian switching under local Lipsehitz condition.In chapter 3, the convergence in mean and in probability between numerical solutions and exact solutions of stochastic differential delay equation with Markovian switching and Poisson jump is discussed under local Lipsehitz condition.In chapter 4, we consider the stability in distribution of neutral stochastic differential delay equation with Markovian switching and Poisson jump by Lipsehitz metric.