The Existence And Uniqueness Of Solutions For Reflected Mean-field Backward(Doubly)Stochastic Differential Equations With Time Delayed Generators

Mean field theory is a crucial theoretical analysis method,which plays a important role in diverse fields such as modern theoretical physics,quantum chemistry,economics and finance,information science and so on.It can simplify the complex problems by approximating all the influences on an object in a random environment to an external field and replacing the sum of single effects with the average effect,thus transform-ing a high-order and multi-dimensional problem that is difficult to be solved into a low-dimensional problem.In 2009,Buckdahn et al.studied a special class of mean-field problems and introduced mean-field backward stochastic differential equations.Since then,the research of backward stochastic differential equations with the mean field framework has been enriched,which provides a powerful tool for the research of stochastic partial differential equations,stochastic optimal control and other fields.This dissertation mainly studies the reflected backward stochastic differential equa-tions with time delayed generators in the framework of mean field theory,that is to say,the reflected mean-field backward stochastic differential equations with time delayed generators is studied.Firstly,the solutions of the equation and its properties are studied under the Lipschitz hypothesis and the terminal condition of square integrability.In more detail,two priori estimates of the solutions are given,the existence and unique-ness theorem is proved by using the fixed point theorem,and the local comparison theorem of process Y on random interval is obtained.Secondly,the reflected mean-field backward stochastic differential equations with time delayed generators is studied in the case where the generators and the terminal value are p(1<p<2)-integrable.It is shown that the priori estimates and the existence and uniqueness theorem of the L~psolutions of the equation are obtained under the condition that the Lipschitz constant and the terminal time are sufficiently small.Finally,the reflected mean-field backward stochastic differential equations with time delayed generators is further generalized.By adding the doubly random term,the reflected mean-field backward doubly stochastic differential equations with time delayed generators is obtained.And the equation is studied under Lipschitz hypothesis and the terminal condition of square integrability,as a consequence,the priori estimates and the existence and uniqueness theorem of solutions are obtained under certain conditions.In this thesis,the reflected mean-field backward stochastic differential equations with time delayed generators and the reflected mean-field backward doubly stochastic differential equations with time delayed generators are studied for the first time,and the existence and uniqueness theorem of solutions and other properties of the equa-tion are proved.It makes up for the vacancy in the theory of mean-field backward stochastic differential equations,and provides theoretical support for the applications of backward stochastic differential equations in the framework of the mean field theory.