The Study Of Jensen's Inequality For Two-dimensional G-expectation

In 1997, Peng introduced a nonlinear mathematical expectation named g-expectation and conditional g-expectation through backward stochastic differential equations, established the theoretical basis of g-expectation under a certain framework,and obtained some basic properties of g-expectation.Then he put forwand the concept of g-martingale on the base of g-expectation. Many scholars have been found that the theory of g-expectation can provide a powerful tool for economic theorys and can lay the theoretical analysis basis of "risk aversion" and "uncertainty aversion" in uncertain market environment for the accurate description. We can give the definition of utility function by g-expectation.Then it offers a good tool for pricing under incomplete financial markets. Scine the concept of g-expectation was put forward, the general linear mathematical expectation has been popularized to non-linear condition, and it can solve many practical problems in the economy.As we all know, when the expected utility function which is defined by mathematical expectation denotes risk aversion and risk preference, the Jensen's inequality of the mathematical expectation plays a key role. And accordingly, g-expectation of Jensen's inequality is also related to the utility function which is defined by g-expectation under uncertain condition. We want to know that if the utility function can describe the uncertain aversion or uncertain preference.In order to apply the theory of g-expectation to solving economic and financial problems better, two-dimensional g-expectation based on Jensen's inequality is studied in this paper.This paper is divided into five chapters. The first chapter introduces the development history of backward stochastic differential equations and research actuality, and describes the definition, property of g-expectation, conditional g-expectation and comparison theorem of BSDE. The second chapter describes the research of one-dimensional g-expectation and conditions of Jensen's inequality which were studied by many scholars. The definition of two-dimensional g-expectation is given in the third chapter. We discuss the necessary and sufficient conditions of Jensen's inequality when the generators meet certain conditions. In chapter four, we apply the Jensen's inequality for g-expectation into the theory of economy, and bring the definition of preference in uncertain enviroment.The fifth chapter is a summary and outlook. We summarize the problems in this paper, and bring forward the deficiencies and the direction which we will continue to research.