As a basic tool researching random phenomenon, the classic mathematical expectation theory, by and large, has been perfect. But with the development of science, a lot of uncertainty phenomenon without additive and repetitive property appears in natural and social sciences. Such phenomenon can not be exactly described by classic mathematical expectation theory as a result of the linearity of the theory. The expectation theory under g-framework based on the BSDE maintains most of the good properties except the linearity, thus can be used as a tool to describe such uncertainty phenomenon precisely, leading to its broad development prospects and application space.This dissertation analyzes the expectation theory under the g-framework uniformly referring the classic mathematical expectation theory, introduces three basic concepts of it: g-expectation, conditional g-expectation and g-martingale, summarizes and introduces some basic properties, inequalities, convergence theorems of the three concepts. On this basis, g-covariance and the moment inequality of g-expectation are introduced and proved. A new proof of the convergence theorem of condition g-expectation is given. In addition, the representation theorem of a special g-expectation denoted byεμunder Lipschitz condition is extended to a special g-expectation denoted byελ(t) under non-Lipschitz condition. Relevant content of the expectation theory under g-framework are combed to make the system more structured and complete. Finally, via some examples, this paper analyzes that the expectation theory under g-framework can be used to describe the uncertainty aversion of the insured and define uncertainty price for the insurance policy in uncertainty environment, and solution some problems of optimal investment decision in ambiguous environment. |