The Study Of A Few Kinds Of Stochastic Differential Equations In The G-Expectation Framework

We are concerned with a few kinds of stochastic differential equations driven by G-Brownian motion and G-Levy process in the G-expectation framework.This thesis,consisting of five parts,is organized as follows:In Chapter 1,the backgrounds of our investigations and some preliminaries are given.In Chapter 2,we study the existence and uniqueness of solution of a class of re-flected backward stochastic differential equations driven by G-Brownian motion(R-BGSDEs).We use a method different from that in[48].Precisely,we take advantage of the G-martingale representation theorem in[76],the optimal stopping under G-expectation framework in[16]and the method presented in[14].However,we also need the G-supermartingale representation theorem which has not been proved in the G-framework,we prove it.Finally,we drive the unique solution of the equation and some estimates.Besides,We obtain the comparison theorem for RBGSDEs which is a powerful tool in the theory of reflected backward stochastic differential equation and will be used in the following chapters.It should be noted that the above equations require generator f to satisfy the Lipschitz condition.In the third part of this chapter,we prove that the RBGSDEs have at least one solution when the generator f does not satisfy the Lipschitz condition,which generalize the results of[48].In Chapter 3,we study the forward-backward stochastic differential equations driven by G-Brownian motion.In the first part,by introducing the iterative algorithm,we prove that there exists a unique global solution for the fully coupled forward-backward stochastic differential equation driven by G-Brownian motion with some monotone coefficients,which generalize the existing conclusions.In the second part,we study the reflected forward-backward stochastic differential equation driven by G-Brownian motion satisfying the obstacle constraint with continuous and Linear growth coefficients.We prove that there exists at least one solution for the equation.One of the difficulties is that the coefficients do not satisfy the Lipschitz condition.On the other hand,the dominated convergence theorem does not hold in G-framework.More-over,the classical Skorohod condition is substituted by a G-martingale condition in the reflected backward stochastic differential equation.We use the smoothing method for continuous function presented in Lepeltier-San Martin[44],the comparison the-orem for forward and reflected backward stochastic differential equations(which has been proved in Chapter 2),the dominated convergence theorem with respect to the integral of time t,the downward monotone convergence theorem in the G-framework,and then construct an approximation sequence.Finally,we prove that the limit of the sequence is a solution of reflected forward-backward stochastic differential equation driven by G-Brownian motion.In Chapter 4,we prove the existence and uniqueness of the mild solution for a class of neutral stochastic partial functional integro-differential equations driven by G-Brownian motion with non-Lipschitz coefficients.Our results are established by means of the Picard approximation.Moreover,we establish the stability of the mild solution.An example is given to illustrate the theory.In Chapter 5,we consider a class of stochastic differential equations driven by G-Levy process.In the firrst part,we prove the BDG-type inequality for G-stochastic cal-culus with respect to G-Levy process.On that basis,the unique solution of stochastic differential equation driven by G-Levy process under non-Lipschitz condition is con-structed.Moreover,we establish the mean square exponential stability and quasi sure exponential stability of the solutions by means of G-Lyapunov function method.An example is presented to illustrate the efficiency of the obtained results.In the second part,the existence theory for the vector valued stochastic differential equations driv-en by G'-Levy process with discontinuous coefficients is established.It is shown that the equations have more than one solution if the coefficient f,g,K are discontinuous functions.The upper and lower solution method is used.