Maximum likelihood estimation of McKean-Vlasov stochastic differential equation and its application

McKean-Vlasov stochastic differential equation is a class of complicated and special equation since the drift term is a function of stochastic process and its distribution. This paper discusses the maximum likelihood estimation of parameters in the drift term through transforming McKean-Vlasov stochastic process into homogeneous one and estimates parameters of the latter to discuss that of McKean-Vlasov equation. Then we build a McKean-Vlasov stochastic model for ion diffusion since ions moved by liquid viscous force and also by coulomb interaction related with ion charged distribution, and simulate the changing trajectory of the ion motion through numerical calculation. Results manifest that the ion motion shows strong random property and has the same tendency for different time intervals, however, the smaller of time lag, the more distinct of wave trajectory observed.