Stability Analysis To Numerical Solutions Of Stochastic Differential Equations With General Decay Rate

Stochastic differential equation first appeared in the beginning of 20th century due to some research on statistical dynamic, but it didn't develop anymore as there were not sufficient mathematical tools to process it then. Until the theory of stochastic processes was fully developed, It? strictly defined stochastic differential equations based on It? calculus in 1951. In the next decades, stochastic differential equation developed rapidly, with perfect applications to many fields, especially on signal processing, stochastic control, financial assets pricing and biological population. However, most of stochastic differential equations cannot be solved analytically, which leads to the research on numerical solutions to them.Among those numerous research topics related to stochastic differential equations, stability is always an important one which attracts plenty of attention. And as in the research in stability, there exist 2 ways: one is to do elementary analysis on how to promise the stability of the solution; the other is to investigate the quality of the stable solution. Now as many new topics such as stochastic stabilization, stochastic bifurcation and stochastic resonance appear, more and more people focus on the latter question. And we concentrate on the asymptotic stable rate of the solutions to stochastic differential equations in this paper.Here, we focus on the general decay rate of numerical solutions to stochastic differential equations. In Chapter 1, we introduce the research background, current research results, future research directions and applications. In Chapter 2, we portrait the problem we want to solve, introduce some preliminary knowledge, especially on the conceptΦ(t )-stability. Chapter 3 gives some sufficient conditions to promise theΦ(t )-stability of the analytical solutions to stochastic differential equations in cases of almost surely and p th moment respectively. We discussed three numerical schemes in Chapter 4, i.e. Euler-Maruyama method, Backward Euler method, stochastic theta method. Several sufficient conditions are given to reproduce theΦ(t )-stability of numerical solutions to stochastic differential equations using the three numerical schemes above.