Stability And Convergence Of Milstein Methods For Two Kinds Of Stochastic Delay Differential Equations

If the status of a system is influenced at a certain time by numerous factors and the influence of each factor which brings to the system has great eventuality, we should consider the influence of the ambient noise in the system. When the developing trends of considered system relates to not only the current status but also the history of the past, it is usually described by stochastic delay differential equation. Nowadays, stochastic delay differential equations have been widely used in the fields of physics, chemistry, bionomics, medical, neural network and auto control and so on.Therefore more and more scholars pay close attention to how to solve the stochastic delay differential equations, however, due to the complex of the system, its analytic solutions are hard to obtain, and it makes numerical solution stochastic delay differential equations especially important. Among them, discussing the astringency and stability of the numerical methods is a significant research subject.In this paper, at first we discuss the convergence of Milstein methods for the scalar Fokker-Planck equations with a noise process. It is proved that the Milstein methods with linear interpolating process for the scalar Fokker-Planck equations are convergent. Second we discussed the numercial stability of neutral delay differential equations. It is proved that the Milstein methods for neutral stochastic delay differential equations are mean-square stable under suitable condition. Moreover the numerical experimentations provided at the end of each chapter prove the theoretical result true.