Stability For Stochastic Functional Impulsive Differential Equations Driven By Brownian Motion And Fractional Brownian Motion

In this paper,the uniqueness and exponential stability of the mild solutions of following stochastic functional impulsive differential equations driven by Brownian motion and fractional Brownian motion are discussed by using the contraction mapping principle,(?) where?(1/2,1).This paper is mainly divided into the following parts.In Chapter 1,the history,research background and research status of stochas-tic differential equations are introduced,and the main research contents of this paper are explained.In Chapter 2,lemmas about stochastic integrals of Brownian motion and fractional Brownian motion is introduced.Concepts,theorems and inequalities to be used are proposed.In Chapter 3,a sufficient conditions for the existence and uniqueness of mild solutions are given.First,the existence and uniqueness of the equation is trans-formed into a fixed point problem in Banach spaces.Then,the existence and uniqueness of mild solutions are proved by using the contraction mapping princi-ple.In Chapter 4,the concept and a sufficient conditions for exponential stability of mild solutions are given.First,the exponential stability problem is transformed into a fixed point problem in Banach spaces.And then the exponential stability of mild solutions is proved by applying the contractive mapping principle.Finally,the application of this paper in delay differential equations is introduced and a concrete example is given.