Qualitative Study On Solutions Of Several Kinds Of Stochastic Differential Equations

In recent years,stochastic differential equation has developed rapidly.The theory of stochastic differential equation is widely used in many fields,such as economy,biology,physics,automation and so on.In this paper,we mainly study the existence of solutions for several kinds of stochastic differential equations.In the first chapter,the origin,research significance and research status of fractional calculus,stochastic differential equation and almost periodic function are introduced.Then the main research contents of this paper are given.Finally,the definitions and theorems used in this paper are given.In Chapter 2,we study the existence of almost periodic solutions for a class of impulsive neutral fractional stochastic differential equations with infinite delay.For this kind of equations,we first obtain the mild solutions by using fractional calculus,operator semigroup and random analysis techniques.Then we prove the existence of the mild solutions by using krasnoselskiis fixed point theorem.On this basis,the existence of almost periodic solutions of this kind of equations is obtained.Finally,a concrete example is given to illustrate the correctness and validity of the conclusion.In Chapter 3,we study the existence and stability theory of the solutions of a class of stochastic impulsive fractional differential equations.For this kind of equations,we first give solutions of the equations by using the fractional calculus theory and the operator semigroup theory,and prove the existence of the solutions of this kind of equations by using the contraction fixed point theorem.Then,based on the continuous dependence of the solutions on the initial condition,the stability of the solutions of this kind of equation is obtained.Finally,according to the random analysis technique and Gronwall inequality,the Ulam-Hyers-Rassias stability of the solutions of this kind of equation is obtained.In Chapter 4,we study the existence of positive solutions for a class of fractional order stochastic impulsive differential equations.Because there are random impulses in this kind of fractional order differential equations,it increases the difficulty of research.In view of the existence of positive solutions for boundary value problems,we mainly use the cone theory to solve,we first give the expression of the solutions of this kind of equation with the help of Green's function.Then we use the technique of random analysis and the concrete properties of Green's function to construct the proper cone and full continuous operator.Finally,we use the fixed point theorem of cone extension and cone compression to get the conclusion that there is at least one positive solution to the boundary value problem of this kind of equation.These results improve the existing relevant conclusions.