The Study Of Jensen's Inequality For G-Expectation

Peng introduces no-linear expectation of integrabel variable (g-expectation)[4], which uses Backward Stochastic Differential Equations (BSDE). Except linearity, g-expectation and conditional g-expectation contain plenty of property of classical expectation and conditional expectation. As we know, when expected utility function which is defined by expectation denotes the aversion and preference of risk, the Jensen's Inequality of expectation takes pivotal function. The Jensen's Inequality for g-expectation attracts many people since the introduction of g-expectation. Accordingly, whether or not reasonable Jensen's Inequality for g-expectation is, which is related to describe uncertain aversion and preference using utility function in uncertain condition. B CHMP[1] constructed a counterexample which points that Jensen's Inequality for g-expectation is not correct by all means. This reminds us that giving appropriate condition to the generator g is necessary and important. The Jensen's Inequality for g-expectations is studied in this paper.This dissertation includes the following four chapters:The first chapter introduces the development background of BSDE and the current research condition of Jensen's Inequality for g-expectation. In the second chapter, based on the research of Jianglong and etc., proving strictly the Jensen's Inequality of function of many variables for g-expectation when g satisfies inferior-homogeneous and inferior-additivity. The third part is the further study of the second one. In this part the necessary condition of the Jensen's Inequality for g-expectation is discussed. In the fourth chapter, the special condition about the Jensen's Inequality is considered. It is that if the function g in partly couplingFBSDE is positive homogeneous, X_T satisfies the Jensen's Inequality for g-expectation.