Backward Stochastic Differential Equations Driven By G-brownian Motion And Related Problems

The sublinear expectation G-expectation,which was firstly introduced by Shige Peng,is a famous nonlinear mathematical expectation.Backward stochastic differential equations driven by G-Brownian motion(G-BSDEs in short)is an important part of the theory of G-expectation.G-BSDE can provide a probabilistic interpretation of a fully nonlinear partial differential equation(PDE in short)and price path-dependent contingent claims in the uncertain volatility model.At present,G-BSDE theory has become one of the hot research directions in the field of stochastic analysis and probability research.The Chapter 1 of this paper is the Introduction,which briefly introduces the basic theory of G-expectation and G-BSDE,and the important results related to them.We also introduce our main work of this paper in Chapter 1.From the Chapter 2 on,the problems in the G-BSDE theory are studied deeply and systematically,and some progress has been made.In Chapter 2,we prove the existence and uniqueness of the solutions to G-BSDE,the comparison theorem and the corresponding nonlinear Feynman-Kac formula under the conditions that the generators satisfy Osgood condition in y and are Lipschitz continuous in z.Firstly,the existence of the solution to G-BSDE is proved by Picard iterative method,and the uniqueness of the solution to G-BSDE is obtained by using a priori estimates of the solution(see Theorem 2.4).Based on the solution to G-BSDE,the convolution method is used to construct the G-BSDE approximation sequence,and the corresponding comparison theorem(see Theorem 2.19)is obtained by the convergence property of the solutions of the approximation equations and the generalized comparison theorem which was proposed by Sun(2020)[136].Finally,the corresponding non-linear Feymann-Kac formula is given(see Theorem 2.21).In Chapter 3,we prove the existence and uniqueness of the solutions to G-BSDE and comparison theorem under the condition that the generators satisfy a weakly monotone and a linear growth condition in y and are Lipschitz continuous in z.First of all,by using the convolution method to build the Lipschitz convolution functions as the generators for the G-BSDE approximation sequence,we obtain the uniform boundedness of estimations for such G-BSDE approximation sequence in the view of the properties of the convolution function,and prove the convergence of solution to G-BSDE approx-imation sequence by the global convergence propertiey between uniformly continuous function and its convolution function,and by the monotone convergence theorem under the capacity theory.Then we get the existence of solution to the G-BSDE.At the same time,an appropriate prior estimates are applied to prove the uniqueness of the solution(see Theorem 3.11).Secondly,based on the above results,the corresponding comparison theorems are obtained by using a similar method in theorem 2.18 in chapter 2(see theorem 3.13).In Chapter 4,we study the existence and uniqueness of solutions to G-BSDE under the condition that the generators are a class of non-Lipschitz continuous and are Lipschitz continuous in z.In classical BSDE theory,Wang-Huang(2009)[144] proposed such condition and obtain the existence and uniqueness of the solutions to BSDE.Under the framework of G-expectation,We still use the Picard iterative approximation method,and then discuss the uniform boundness of solution to the approximation equation and a priori estimates of the convergence on interval [T1,T ],and finally prove the existence and uniqueness solution to G-BSDE on interval [0,T] by interval backward recursion method(see Theorem 4.8).In Chapter 5,we obtain the existence and uniqueness of solution to G-BSDE on the finite interval [0,T] under the condition that the generators satisfy uniformly continuous in y varying with time t and are Lipschitz continuous in z satisfying inconsistency with time t.Firstly,the convolution technique is used to construct the upper and lower GBSDE approximation sequence,and under the condition of linear growth varying with time t for the generator,the uniform boundness of the solution((?)~n,(?)~n,(?)~n)of the approximation equations,and a prior estimates of the convergence of (?)~n are obtained.Secondly,we build the approximation equations by Picard iteration for the above two kinds of upper and lower G-BSDE approximation sequence,and apply ODE method to control the difference of the solutions (?)~n-(?)~n by using the linearization technique in Hu-Qu-Wang(2020)[54] to the Picard iteration approximation equations.Finally,the existence and uniqueness of the solutions to G-BSDEs are proved by using the technique of G-stochastic analysis(see Theorem 5.20).Based on the existence and uniqueness theorem of solutions,the comparison theorem is obtained by using comparison theorem under thecondition of Lipschitz's continuous varying time(see Theorem 5.23).