Existence,Uniqueness And Estimate Of Solutions Of Stochastic Functional Differential Equations Driven By G-Brownian Motion

There are stochastic phenomena widely in natural science,engineering technology,financial management and other disciplines,including our daily life.With the devel-opment of the theory and applications of stochastic differential dynamical systems,the existence,uniqueness,moment estimation and stability of the solutions of stochastic functional differential equations have aroused great interests of scholars.To our knowl-edge,the theory and applications of stochastic differential equations are mostly driven by Brownian motion.As we know that the G-Brownian motion,which nontrivially generalizes the classical Brownian motion,has new structure,rich theory and powerful applications.Because of the extensive application of G-Brownian motion,the theo-ry and application of stochastic differential equations driven by G-Brownian motion have aroused the great concern of scholars.This dissertation is concerned with the existence,uniqueness and estimation of solutions for several stochastic functional dif-ferential equations driven by G-Brownian motion.The main work of this dissertation is as follows.The research background and significance of this subject are summarized in Chap-ter one.The preliminaries and lemmas are also given,including the basic theory of stochastic differential equations,the concept of G-Brownian motion and some impor-tant stochastic inequalities.The second chapter is concerned with stochastic neutral functional differential equa,tions driven by G-Brownian motion with infinite delay.The basic knowledge of stochastic neutral functional differential equations driven by G-Brownian motion with infinite delay is given.After establishing the Picard approximation sequence,the convergence of the Picard approximation sequence is obtained by Holder inequality,Gronwall inequality and the properties of the G-Brownian motion.Thus,the existence and uniqueness of the solutions are obtained.Finally,the several moment estimations of solutions are received by using the analysis technique of stochastic inequality.The stochastic neutral functional differential equations with finite delay driven by the finite G-Brownian motions are discussed in Chapter 3.The basic knowledge of the finite G-Brownian motions is introduced in detail.For this kind of stochastic neutral functional differential equations,under the linear Growth and Lipchitz conditions of every G-Brownian Motion,a specific Picard approximation sequence is constructed,the existence and uniqueness of solutions are obtained by the Holder inequality and the analysis technique,also the moment estimations of solutions are investigated.Chapter four is concerned with a kind of the stochastic neutral functional dif-ferential equations with pantograph delay driven by the G-Brownian motion.As the pantograph delay is a kind of infinite delays,the special technique of amplification and compression methods is employed to deal with pantograph delay.The existence,uniqueness and estimate of solutions are obtained by using of Picard approximation sequence,Gronwall inequality,Holder inequality and Burkholder-Davis-Gundy inequal-ity.Chapter 5 focuses on the stochastic functional differential equations with piecewise constant delay driven by the G-Brownian motion.The SFDEs of retarded type and neutral type are studied,respectively.By using of fundamental theory of stochastic differential equations combined with analysis technique,the existence,uniqueness and estimate of solutions are obtained.As applications of our main results,the accurate representations and the estimation of solutions are given for linear retarded and neutral equations,respectively.