The Existence And Stability Of Stochastic Differential Equations And Its Applications

In the real world,there always exist stochastic perturbations and impulse phenomena.In order to more accurately describe the actual situation,the modeling of impulsive stochastic differential equations is more practical.Therefore,this dissertation mainly investigates the existence and stability of mild solutions for two types of impulsive stochastic differential equations,and then we consider epidemic models with stochastic perturbation and analyze the dynamics of stochastic SIRQ epidemic model with isolation measures.The whole dissertation consists of the following four chapters.In chapter 1,firstly,we introduce the history of stochastic differential equations,the theory of stochastic stability and the research progress of stochastic epidemic model;secondly,the main content of the dissertation is given;finally,the preliminary knowledge of our works is introduced.Chapter 2 is devoted to the existence and stability theory for a class of impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup.Under the noncompactness assumption of operator semigroup,we derive an inequality of noncompact measures for Ito integral and the existence of solutions for this equations are established by Hausdorff measure of noncompactness and Monch's fixed point theorem.Then,the Ulam-Hyers-Rassias stability is solved by using stochastic analysis techniques and Gronwall inequalities.Finally,the exponential stability of solutions are obtained by constructing a new impulsive integral inequality.The correctness and the effectiveness of the conclusions are verified by an example.Chapter 3 focus on the existence and stability theory for a class of stochastic differential equations with random impulses.The existence theorem is obtained by using Lipschitz conditions and Krasnoselskii's fixed point theorem.Then,the stability of the equations through the continuous dependence of solutions on initial condition.Finally,by employing stochastic analysis techniques and Gronwall inequalities,we prove the Ulam-Hyers-Rassias stability of solutions to such equations.As the main application of stochastic differential equations,the chapter 4 is concerned with the stochastic SIRQ epidemic model with isolation measures.in this chapter,we first prove the global existence and uniqueness of positive solution of stochastic model,and we conclude that the disappearance and prevalence of diseases through the study of stability and instability of solution for stochastic model.Finally,by constructing the appropriate Lyapunov function,the asymptotic behavior of the solution of the stochastic model in the corresponding deterministic model around the endemic proportion equilibrium is analyzed.Numerical simulations are carried out to illustrate the correctness and the effectiveness of the conclusions.