The Properties Of Solutions To McKean-Vlasov Stochastic Differential Equations With Non-Lipschitz Coefficients

Stochastic differential equations play an important role in stochastic analysis.They not only have many extensive applications in stochastic system control,filtering theory and engineering structure analysis,but also have unexpected connections with other branches of mathematics,such as measurement theory and partial differential equations.In 1902,when Gibbs studied statistical mechanics,he put forward the problem of integrals of differential systems under the condition that the initial state was random.Later,Ito first discussed the problems of stochastic differential equations in 1951.For half of a century,scholars have conducted extensive and deep theoretical research on stochastic differential equations.Looking back on the past research in recent years,mathematicians have been working on the properties of the solutions,but for different types of stochastic differential equations,the properties of the solutions are quite different.Before the 1990s,the theory of stochastic differential equations driven by the Brown motion occupied an important position in stochastic analysis,which was ap-plied in finance,stochastic networks and other fields.However,in recent years,the mathematicians have gradually changed the research subject to McKean-Vlasov stochastic differential equations,whose drift coefficients and diffusion coefficients depend on the processes themselves and their prob-ability distributions.McKean-Vlasov stochastic differential equations have pratical applications in fields such as interacting particle systems and game theory.In this paper we mainly study the well-posedness and stability of solutions to a type of s-tochastic McKean-Vlasov equations with non-Lipschitz coefficients.On one hand,by an Euler-Maruyama approximation,the existence of weak solutions is proved,and then we discuss the uniqueness in the sense of probability law of the weak solutions.Since the uniqueness in the sense of probability law does not imply pathwise uniqueness,we observe pathwise uniqueness of weak solutions under some conditions.Finally,it is shown that the Euler-Maruyama approximation has an optimal strong convergence rate.On the other hand,first,sufficient conditions are given for the exponential stability of the second moments for their solutions in terms of Lyapunov functions.Then we weaken the conditions and furthermore obtain exponentially 2-ultimate boundedness of their solutions.At last,the almost surely asymptotic stability of their solutions is proved.