Application Of The Convergence Of Random Differential Equations And The Taylor Expansion Method

Generally, explicit solutions can hardly be obtained for stochastic differential equations.Even when such a solution can be found,it may be only in implicit form or too complicated to visualize and evaluate numerically.So, several numerical schemes such as Euler- Maruyama method ,backward Euler-Maruyama,θmethod, stochastic Taylor expansion e.t. have been developed to produce approximate solutions to study relations between the stability of numerical solutions and true solutions and to investigate all kinds of convergences between numerical solutions and true solutions.However,most literatures have studied the convergence between numerical solutions and true solutions of stochastic differential equations under the global or local Lipschitz condition. Sometimes,the global or local Lipschitz condition is so strong that the drift coefficient and diffusion coefficient of many stochastic differential equations can not satisfy.This brings about difficulties to investigate the convergence of between numerical solutions and true solutions. As far as I am concerned, there is few papers concerning with non-Lipschitz condition.In this paper, utilizing the Euler-Maruyama method, we investigate the convergences between numerical solution and true solution of stochastic differential delay equations with Markovian switching and Poisson jump in the L~l and L~p senses. Furthermore, by means of different numerical schemes, the rate of convergence between numerical solutions and true solutions is different. In this article, we also employ Taylor expansion method to consider the convergence between numerical solutions and exact solutions of stochastic differential delay equations with Markovian switching in If sense. Through the process of proof, we can see the rate of convergence between numerical solutions and true solutions is faster than the Euler-Maruyama method. Therefore, the Taylor expansion method is superior to the Euler-Maruyama method.The paper contains four chapters.Chapter 1 is the preface.In chapter 2, we investigate the convergence between numerical solutions and exact solutions of stochastic differential delay equation with Markovian switching and Poisson jump under non-Lipschitz condition in the L~l and L~p senses.In chapter 3, the convergence between numerical solutions and exact solutions of stochastic differential delay equation with Markovian switching is discussed under local Lipschitz condition by Taylor expansion method.