Backward Stochastic Differential Equations Under Nonlinear Expectations

In 1990,Pardoux and Peng[65]originally introduced nonlinear BSDEswhere the generator f is Lipschitz continuous with respect to(y,z)and the terminal variable ζ is square integrable.Based on this pioneering work,a lot of works to weaken the Lipschitz condition on the driving generators and the integrability condition on the terminal variables have been done(see[1,4,5,10,16,20,21,26,42,46,49,63]).In 2009,Buckdahn,Djehiche,Li and Peng[6]obtained a new type of backward stochastic differential equations:mean-field backward stochastic differential equations,when they studied a mean-field problem by purely stochastic method.Later,Buckdahn,Li and Peng[7]deeply investigated mean-field BSDEs under Lipschitz condition.The theory of BSDEs and the theory of mean-field BSDEs have been developed very fast and applied in in various fields,such as partial differential equations,stochastic control and stochastic differential games,numerical analysis,mathematical finance(see[8,15,17,24,25,27,28,29,30,45,52,64,66,67,68]).Peng[69]defined a nonlinear-expectation:g-expectation.Research shows that the theory of g-expectation is a powerful tool for studying probabilistic uncertainty mod-els,where the involved probability measures are absolutely continuous about Wiener measure.However,Volatility uncertainty model in many financial problems involve a family of mutually singular probability measures.Motivated by this question,Peng[72]established G-expectation theory via the following nonlinear heat equations:where G(·):Sd→R is a given monotonic and sublinear function.Recently,the theory of G-expectation has been extensively applied to asset pricing theory,stochastic optim al control and differential utility model under volatility ambiguity(see[14,18,19,34,35,56]).Furthermore,Peng[70]and Gao[22]studied the existence and uniqueness of solu-tions to G-SDEs under Lipschitz condition.Lin and Bai[2]studied G-SDEs under inte-gral Lipschitz condition.Different from the classical martingale representation theorem,there is an additional decreasing G-martingale term K in G-martingale representation theorem,so that the solution of G-BSDEs is a triple of processes(Y,Z,K):Hu,Ji,Peng and Song[32,33]investigated the existence and uniqueness of solutions to G-BSDEs under Lipschitz condition,its comparison theorem and a nonlinear Feynman-Kac formula.Hu,Lin and Soumana Hima[41]studied the existence and uniqueness of the solutions to G-BSDEs under quadratic growth conditions.Wang and Zheng[78]studied G-BSDEs,whose generators are Lipschitz continuous in y and uniformly continuous in z.For more developments on the theory of G-SDEs and G-BSDEs,one can refer to[36,37,38,51,53,55,57,58,61,62,78].This thesis is mainly devoted to G-BSDEs under non-Lipschitz condition,mean-field G-SDEs and mean-field G-BSDEs under lipschitz condition,mean-field G-BSDEs under non-Lipschitz condition.Now we give the introduction of the main content of this paper.In Chapter 1,we review some basic notions and properties of G-Brownian motion,G-Ito integral,and G-BSDEs.In Chapter 2,we study G-BSDEs with non-Lipschitz coefficients in y.We con-struct local solutions by performing Picard iteration on the Y-term,utilizing a prior estimates for solutions of G-BSDEs under Lipschitz condition and the maximum in-equality for G-martingale.By the method of backward iteration of local solutions,we get the global solution.The unqiueness of solutions can be obtained via G-Ito formula.In addition,we give a priori estimates for solutions and comparison theorem of G-BSDEs.Finally,we get representation theorem for generators and converse comparison theorem of G-BSDEs by a priori estimates for solutions.In Chapter 3,we further study G-BSDEs with uniformly continuous coefficients in(y,z).First,we construct two approximating sequences of G-BSDEs under Lipschitz condition and derive a uniform estimate for the their solutions by utilizing a linearization method of G-BSDEs,then the existence and uniqueness of solutions is proved.Next,we obtain comparison theorem,representation theorem for generators of G-BSDEs through a limit argument,and then obtain a converse comparison theorem.Finally,we give the relationship between G-BSDEs and the viscosity solutions of a fully nonlinear parabolic equation.In Chapter 4,we introduce mean-field G-SDEs and mean-field G-BSDEs.Due to the appearance of G-expectation E in the mean field G-BSDEs,we need to limit the generators to the space MGp(0,T).This is different from G-BSDEs.Unlike the classic BSDEs,the appearance of the G’-martingale term makes it impossible for us to construct a contracting map involving Z.The fixed point principle cannot be applied to solve the mean field G-BSDEs.Through a contraction argument for Y-term in small intervals and a method of backward iteration,we get the unique solution of the mean-field G-BSDEs.Then,we give comparison theorem and converse comparison theorem of mean-field G-BSDEs,and a probabilistic interpretation for a fully nonlinear partial differential equation of mean-field type under G-expectation framework.Finally,we investigate the stochastic differential utility model of mean-field type under G-expectation framework.In Chapter 5,based on chapters 2,3 and 4,we further study mean field G-BSDEs under non-Lipschitz condition,and obtain the existence and uniqueness theorem and comparison theorem.