Stochastic Mckean-vlasov Equations Driven By Continuous Martingales

It is well known that the limiting McKean-Vlasov process has many interesting asymp-totic behaviors. Therefore, as a useful mathematical tool for the process, McKean-Vlasovequations have attracted many scholars’ research attention. The Mckean-Vlasov equationsare also the research topic of extensive focus both in the field of statistical physics and in thearea of stochastic analysis, which are of great theoretical significance and practical value.The driving process of classical stochastic McKean-Vlasov equations is Bronwnianmotion. However, the special characters of Brownian motion put restrictions on the ap-plications of stochastic McKean-Vlasov equations in practice. As in the basic theory ofstochastic differential equations, this paper generalizes the classical theory of stochasticMcKean-Vlasov equations under the frame of the continuous martingales. The method ofrandom time-change is used to change the two stochastic differential equations under in-vestigation into another two equations with better properties. The equivalent martingaleproblem is established under proper assumptions. And the equivalence of the solution to themartingale problem and that to the stochastic McKean-Vlasov equation is used to prove theexistence of the weak solution to the stochastic McKean-Vlasov equation.This paper is organized as follows. Chapter one is introduction, which introduces thebackground, significance and status of research of McKean-Vlasov equations as well asthoughts of research and main contents. The preparation knowledge is listed in Chaptertwo. Chapter three introduces the form of the equation that the interacting diffusionprocesses satisfy and the assumptions that coefficients satisfy and establishes the momentestimates and the tightness of the distribution and their proofs. Chapter four is the corecontent of this paper, in which the equivalence of the solution to the martingale problemand that to the stochastic McKean-Vlasov equation is used to discuss the existence ofweak solution. The pathwise uniqueness and the existence and uniqueness of strongsolution as well as a propagation of chaos result are also obtained in this chapter. Final-ly, Some areas that can be improved and further investigated are given in the conclusion part.