Existence Of Periodic Solutions In Distribution For Several Classes Of Stochastic Differential Equations

The existence of periodic solutions is a subject of great concern in the theory of differential equations.For stochastic differential equations,the existence of periodic solutions is a challenging problem.The study of periodic solutions of such equations has attracted the attention of many scholars.Different from the periodic solutions of deterministic differential equations,there are two kinds of periodic solutions considered in stochastic cases.One is pathwise random periodic solutions,which is a kind of local periodicity,and the other is periodic solutions in distribution,which is a kind of global periodicity.The latter is considered in this dissertation.The periodic solution in distribution is observed from a statistical point of view.Since there is a Brownian noise in the dynamics,there is no hope to have periodicity for the state itself.But the distribution of the state has a deterministic evolution,for which one can hope to have periodicity.Therefore,we consider the existence of periodic solutions in distribution for stochastic functional differential equations with jumps,mean-field stochastic differential equations and infinite dimensional stochastic differential equations-stochastic lattice differential equations.This dissertation is divided into five chapters as followings:Chapter 1 is the introduction,including the development and research status of stochastic functional differential equations,periodic solutions in distribution,mean-field theory and stochastic lattice systems.Moreover,the main results and arrangements of this dissertation are introduced.Chapter 2 recalls some preliminary knowledge needed in this dissertation.Chapter 3 discusses the existence of periodic solutions in distribution of stochastic functional differential equations with jumps.Firstly,the comparison principle of stochastic functional differential equations with jumps is proved based on viscosity solutions and viability.Then a monotone sequence is obtained by using the method of upper and lower solutions and the comparison principle.Consequently,the existence of periodic solutions in distribution for stochastic functional differential equations with jumps is proved.Chapter 4 studies the existence of periodic solutions in distribution of meanfield stochastic differential equations.We give two definitions of upper and lower solutions of mean-field stochastic differential equations,and prove the existence of periodic solutions in distribution of mean-field stochastic differential equations by using the comparison principle and Banach’s contraction principle.Moreover,we give examples of weakly interacting particle system and Black-Scholes model as applications of our conclusions.Chapter 5 focuses on the existence of periodic solutions in distribution of stochastic lattice differential equations.First,we study the well-posedness of solutions of stochastic lattice differential equations.Then,by using upper and lower solutions and the existence of a contraction mapping,a monotone structure is obtained.Furthermore,the existence of periodic solutions in distribution of stochastic lattice differential equations is proved.