Convergent Theorems For The Generalized G-expectation

In this paper,some properties of BSDEs on infinite horizon and g-expectations are studied,and then the Levi,Fatou and Lebesgue convergent theorems of the generalized g-expectations are verified.Backward stochastic differential equations,BSDE's in short,were first introduced by Pardoux and Peng[9].The solution of the SDE turns the assured original condition to the future situation which is usually uncertain to investigate the statistical rule;but the solution of the BSDE turns the uncertain terminal condition to the assured original condition to constitute the scheme.It has been since widely recognized that they provide a useful framework for formulating many problems in mathematical finance; see in particular[10]and[2].They also appear to be useful for problems in stochastic control and differential games(see[2]and[14]),for constructing -martingales on manifolds with prescribed limits(see[16])and providing probabilistic formulas for solutions of systems of quasi-linear partial differential equations(see[13]).The definition of g-expectation and conditional g-expectation were first introduced by Peng[1],and then some properties were studied by Peng,Briand and Coquet etc.,and successfully extend them to nonlinear expectation.To generalize the BSDE theorem,Chen and Wang[7]verified the existence and uniqueness of the solution of BSDE on infinite horizon,and gave new definition of g-expectation through the solution.Chen[6]used another way:operator extension to extend the domain of g-expectation from the space of square integrable random variables to the space of integrable random variables.Combination of these two results,in a similar way Qian Jingjing,Huang Zhen,Wang Xiangrong[8]extended the definition of g-expectation in[6]to infinite horizon.The comparison theorem for real-valued BSDEs turns out to be one of the classic results of this theory.It is due to S.Peng[12]and then generalized by Pardoux-Peng [13]and El Karoui-Peng-Quenez[2].It allows to compare the solutions of two real-valued BSDEs whenever we can compare the terminal conditions and the generators.In this paper,the comparison theorem on infinite horizon is verified similar to the way of Peng.With the basis of the contents above,this paper gives the definition of the generalized g-expectation,and some properties of which are studied,and this paper also verifies Levi,Fatou and Lebesgue convergent theorems with these properties and those proof methods of the classic convergent theorems.