Characters Of Non-linear Expectation

There are many uncertainties in nature society and social society that can't be measured by linear expectation. Such as how to measure people's preference in uncertain environment, a common method is using utility function. But as Allais disoblige and Ellsderg disoblige were raised,this method was challenged. Linearity of classic expectation was one of most important reason of these disobliges,so experts tried to deal with these problems with non-linear expectation.From 90's last century,g-expectation based on BSDE and it's characters had been developed quickly, these results solved many questions in different fields. A famous model in financial called Black-Schoies formula is the answer of a BSDE. It was very important in assets pricing, and was used for calculating the price of assets. And g-expectation based on BSDE and it's characters are used more widely.This article introduces the characters of condition expectation which was used for dynamic models. Such as,when g is superhomogeneous g-expectation satisfies Jensen's inequality;when F-expectation is superhomogeneous and constant additive,F-expectation satisfies Jensen's inequality too;when g is linear with z or satisfies the property of comonotic additivity,g-expectation can be represented by Choquet integral.In addition,this article dicussed condition F-expectation's property of comonotic additivity. Jiang(2006,[19]) has proved that condition g-expectation satisfies the property of comonotic additivity when g-expectation satisfies the same property and is subadditive or supersdditive. In this article,I reduce the proof of the theorem. And extend it to condition F-expectation,prove that condition F-expectation satisfies the property of comonotic additivity when F-expectation satisfies the same property of and is subadditive or supersdditive. This property is useful in financial.It means the prices of contingent claim satisfy the property at time t before T when they satisfy the property at time 0. In addition this article raises a new idea about defining condition expectation of risk premium H(X,α) defined in Wang(2000,[30]).