Stability Analysis Of Predictor-Corrector Methods For Stochastic Delay Differential Equations

As a fundamental question in computational mathematics,the bidirectional infer-ence of the stability between analytical solution and numerical solution of the equation is still an open problem.This paper focuses on whether the numerical algorithm can preserve the stability of the original problem to the greatest extent,and addresses the following two questions:(?)If the equation is stable,whether the numerical method can reproduce this stability of the exact solution?(?)Under the same stable conditions,for what choice of time step size does the numerical method infer the stability of the exact solution?The predominant work is summarized as follows:In chapter one,the origin of stochastic delay differential equations and the pio-neering work of scholars are given,significance of this topic and research purpose and methods are explained.In chapter two,we provide the elementary knowledge leading to the following chapters.In chapter three,we construct a class of Predictor-Corrector Methods,discuss the delay-dependent stability of linear stochastic delay differential equations,and prove the equivalence theorem for asymptotically mean square stability between the numerical solutions and the analytical solutions with two view points.In chapter four,we investigate the relationships of stability regions between the numerical solutions and the analytical solutions,and further give the explicit formula in terms of the time step size and the implicit degree parameter,which can adjust the size of stability region.In chapter five,numerical experiences are showed,the distribution of characteristic roots and the comparisons between the numerical and analytical stable regions are given,thus we verify all the theorems of this paper.In chapter six,we summarize the contributions and point out the further research direction.