Research On Generalized G-Expectations And Releated Problems

By Pardoux and Peng (1990), we know that there exists a unique adapted and square integrable solution to a backward stochastic differential equation (BSDE for short) of type providing that the function g is Lipschitz in both variables y and z, and that ζ and the process (g(t,0,0))t ∈[0,T] are square integrable. We denote the unique solution of BSDE Since then, many researchers have been working on this subject and related proper-ties of the solutions of BSDEs, due to the connection of this subject with mathematical finance, stochastic control, partial differential equation, stochastic game and stochastic geometry and mathematical economics. In recent years, people are more and more interested in Lp (1<p<2) solutions of BSDEs and related problems.The dissertation is divided into five chapters.In Chapter Ⅰ, we study the existence and uniqueness theorem for Lp solutions to a class of infinite time interval BSDEs. Furthermore, we introduce generalized g-expectations and generalized g-martingales via the Lp solutions and prove the stability theorem of generalized g-expectations.In Chapter Ⅱ, we obtain a comonotonic theorem for BSDEs in Lp. As applica-tions of this theorem, we study the relation between Choquet expectations and min-imax expectations and the relation between Choquet expectations and generalized g-expectations.In Chapter Ⅲ, by using LP weak convergence method on BSDEs, we obtain "the limit theorem of g-supersolutions. As applications of this theorem, we study the de-composition theorem of εg-supermartingale, the nonlinear decomposition theorem of Doob-Meyer’s type and so on. Furthermore, by using the decomposition theorem of εg-supermartingale, we provide some useful characterizations of εg-evaluation by the generating function g without the assumption that g is continuous with respect to t.In Chapter IV, we introduce the notion of Ft-consistent expectation defined on L(Ω, FT, P) and prove an existence and uniqueness theorem for solutions and a com-parison theorem of BSDE under εβ-dominated F-expectations. Furthermore, as an application of this comparison theorem, we obtain the decomposition theorem for ε-supermartingales.In Chapter V, we investigate the n-dimensional Jensen’s inequality, Holder’s in-equality and Minkowski’s inequality for dynamically consistent nonlinear evaluations in L1. Furthermore, we give four equivalent conditions on the n-dimensional Jensen’s inequality for εg-evaluations induced by BSDEs in LP. At last, we give a sufficien-t condition on g under which Holder’s inequality and Minkowski’s inequality for the corresponding εg-evaluation hold true.