The Stability Of Neutral Stochastic Differential Equations

From the Brownian Motion was founded to the establishment of the stochastic differential equation(SDE), it took the efforts of Brown, Einstein, Wiener, ?Ito, et al. At present, people increase attention to the theory of stochastic differential equation because it important applications of several disciplines in engineering and science. The stability of stochastic differential equation is an important research direction. We successfully get some results about the stability of Stochastic differential equations by applying Lyapunov function drectly, but we also faced many difficulties at the same time. Luo J W researched Neutral stochastic differential delay equations and got some great results through using the fixed point theory. Now this method has been gradually getting people’s attention and recognition.In this paper, we study the stability of several types of neutral stochastic delay differential equations by using the Banach fixed point theory in the complete space. This topic is mainly divided into five parts, the first part mainly introduces the origin, development, status and a series related background of stochastic differential equations,it also gives a brief introduction of some concepts and basic knowledge used in this paper. The second part discusses the mean square asymptotically stability of a class of non-linear neutral stochastic differential equations through constructing a suitable compression operator, using the relevanted basic knowledge of the second part, we get the conditions ensure this kind of neutral stochastic system with time-varying delay to be mean square asymptotically stable. In the third part, we research the mean square asymptotically stability of a class of nonlinear neutral stochastic differential equations with variable delay by using the fixed point theory and the same method; The fourth part gives the conditions ensure one kind of neutral stochastic system with time-varying delay to be asymptotic p stable and exponential p stable by using fixed point theory. The fifth part gives the conditions ensure one kind of non-linear neutral stochastic system with time-varying delay to be asymptotic p stable and exponential p stable by using the same method.