Estimation And Discrete Approximation Of McKean-Vlasov Stochastic Differential Equations

McKean-Vlasov(distribution-dependent or mean-field)stochastic differential equations(SDEs),which were first proposed by Henry P.McKean in 1966,describe the evolution of particle systems disturbed by noise,and are widely used in biology,finance and other fields.The difference between McKean-Vlasov SDEs and general SDEs is that McKean-Vlasov SDEs depend on the location and probability distribution of these particles.In this paper,we consider the estimation of McKean-Vlasov SDEs.On the one hand,we consider the maximum likelihood estimation of unknown parameters in a class of path-dependent McKean-Vlasov SDEs.The parameter estimation of SDEs is always concerned.After years of research,the methods and techniques tend to be mature and diverse.However,due to the distribution in drift coefficients and diffusion coefficients of McKean-Vlasov SDEs,we can not directly use the traditional methods to estimate the parameters.In this paper,we first prove the existence and uniqueness of solutions for this kind of path-dependent McKean-Vlasov SDEs under non-Lipschitz conditions.Then,we construct the maximum likelihood estimators of unknown parameters and discuss their strong consistency.Then,we give a discrete approximation method for the path-dependent McKean-Vlasov SDEs,and calculate the error between the solution of this equation and that of the discrete approximation equation.On the other hand,we consider the nonlinear filtering problem of McKean-Vlasov SDEs system with correlated noises.The filtering problem is to estimate and predict the unobservable process through the observation process in the stochastic system.The nonlinear filtering problem is solved by Zakai equation and Kushner-Stratonovich equation.At present,the result about nonlinear filtering of McKean-Vlasov SDEs is seldom.In this paper,we first establish Kushner-Stratonovich equation,Zakai equation and distribution-dependent Zakai equation of McKean-Vlasov SDEs system with correlated noises.Then,the pathwise uniqueness,uniqueness in joint law and uniqueness in law of the weak solutions for the distribution-dependent Zakai equation are shown.Finally,the relationship between the distribution-dependent Zakai equation and the distribution-dependent Fokker-Planck equation is studied.In addition,the existence of weak solutions for the Fokker-Planck equation under certain conditions is obtained.