Jensen's Inequality Of G-expectation In General Filtration

Backward stochastic differential equation(BSDE)is widely used in finance and economy.It is an important basic theory to study futures pricing,stochas-tic control and stochastic game.Pardoux and Peng[1]proved the existence and uniqueness theorem of BSDE solution.The theorem is proposed in the framework of natural filtration and is widely used in the study of dynamic risk measurement.However,due to the incompleteness of the financial market,the classical BSDE theory has encountered new challenges.Therefore,it is of great theoretical and practical significance to study BSDE under general filtration.This article first introduces the definition and comparison theorem of the trans-position solution,and briefly describes the properties of g-expectation and con-dition g-expectation and On the super-homogeneity of generator function g with respect to z.based on the above research,it is concluded that if and only if the generator function g is independent of y and is super-homogeneous in z,Jensen' s inequality for g-expectation and condition g-expectation in general fil-tration hold.Secondly,the existence of BSDE transposition solution under the condition of non-Lipschitz is further discussed.Because of the special form of transposition solution,Ito formula can not be used,therefore,according to the existence of transposition solution under Lipschitz condition,Bihari's inequal-ity,Jensen's inequality,Holder inequality,etc.,we proved the boundedness of the sequence of transposition solution under Lipschitz condition.Next,in order to discuss the convergence of transpose solution under the non Lipschitz condition(here in after referred to as the non Lipschitz condition of Mao)given by Mao,by constructing a transpose solution sequence of BSDE,it is proved that the constructed transpose solution sequence is convergent.