The Study Of The Quality Property On Neutral Stochastic Differential Equations

This master thesis consists of three parts, it mainly discusses the existence and uniqueness for the solution of neutral stochastic differential equations with time-varying delay and the estimate for the error between the approximate solution and the accurate solution; By establishing two coupled delay-integral inequalities, the exponential stability for the neutral stochastic linear differential equations with time-varying delay and the neutral stochastic nonlinear differential equations with time-varying delay are discussed, respectively. By using Borel-cantelli Lemma, the almost sure exponential stability for such two systems is also implied.In chapter one, the background and the development of the theory and the application for the neutral stochastic differential equations are introduced. The novelty and the main results in this thesis are given..In chapter two, by constructing a new iterative scheme, the existence and uniqueness for the solution of the neutral stochastic differential equations with time-varying delay can be directly obtained only under the Lipschitz condition and the contractive condition. The estimate for the error between the approximate solution and the accurate solution of such equation is also provided.In chapter three, by establishing two coupled delay-integral inequalities and using the stochastic analytic technique, two exponential criteria on exponential stability and almost sure exponential stability in mean square for the neutral stochastic linear differential equations with time-varying delay and the neutral stochastic nonlinear differential equations with time-varying delay are considered, respectively. Finally, two numerical simulations are conducted.