The Existence And Uniqueness Of The Solutions Of Stochastic Differential Equations With Non-lipchitzian Coefficients

Stochastic differential equations(SDEs)have significant application in many fields such as statistical physics,control theory,population genetics,circuit theory,financial options,etc.When we research a class of SDEs,the primary subject is to figure out the exis-tence and uniqueness of its solution(including the explosion properties of its solutions).People have defined several kinds of solutions of SDEs and several kinds of uniqueness of the solution.The existence and uniqueness of the solution of a SDE with global or local Lipschitz coefficients have been figured out by predecessors,But the Lipschitz continuity is a very "strong" condition.Consequently,when we work on realistic prob-lems we are not always faced with the SDEs with Lipschitz coefficients.So the SDEs with non-Lipschitz coefficients are the issue which raises much concern in stochastic analysis field.This thesis will study the existence,uniqueness and explosion properties of SDEs with non-Lipschitz coefficients.The structure of this thesis is arranged as follows:chapter 1 serves as the introduc-tion of this paper.In chapter 2,we introduce the definition and properties of stochastic integral of Brown motion briefly,as a foreshadowing for us to raise the form of SDEs.In chapter 3,we give the form of the SDEs which we focus on,defining the weak and strong solutions of SDEs,several kinds of uniqueness of their solutions as well as the explosion time of a solution.In chapter 4,We summarize many important results about the existence,uniqueness and explosion properties of SDEs with non-Lipschitz coef-ficients and sort out them.In chapter 5,We present some important results about the existence and uniqueness of SDEs with non-Lipschitz coefficients.In chapter 6,we first review three important results which we have introduced in chapter 4 and chapter 5.we generalize and improve them slightly,as well as providing the proofs of them.Then,on the basis of predecessors' works and methods,we originally propose two new theorems which can contribute to solve the uniqueness of solution of two classes of SDEs with non-Lipschitz coefficients and prove them.