On Representation Of Generalized G-Expectation

Since the seminal work of Artzner, Ph., F. Delbaen, J.-M. Eber and D. Heath[1],[2], the study on coherent risk measures and sub-linear expectations has achieved splendid progress. Roughly speak-ing, a sublinear expectation is a sublinear functional on a certain ran-dom variable space (?), i.e. a functional E on some space such that E[X+Y]≤E[X]+E[Y] and E[λX]=λE[X] forλ≥0,X,Y∈(?). The concept of coherent risk measures arises from the practical area of finance and it has a bright future in application.An important case of sublinear expectation is G-expectation and generalized G-expectation, which was first introduced by Peng [14]. In these papers, Peng exstracted the most essential features of dis-tributions, independence, etc. from classical probability theory and successfully generalized them to sublinear framework. According to the representation theorem of sublinear expectations, G-expecatation can be represented as an upper expectation. Our problem is how to construct a family (?) of probabilities on C[0,∞) such that the G-expectation The paper also studies some properties of upper expectation and its relations with capacity. This article is organized as follows. In section one, we give ba-sic elements about nonlinear expectations and some useful properties of upper expectations. Its relation with Capacity is also given in this section. In section two, we have a brief review of the concepts of the G-framework, including distributions, independence, G-distribution and Generalized G-Brownian Motion. In section three, we begin our con-struction work of a family of probabilities to represent EG and we will prove its tightness.