The Related Properties Between The Generator G Of BSDEs And G-Expectations

In 1990, Pardoux and Peng introduced the following backward stochasticdifferential equation (BSDE):and proved there exists unique adapted solution to this equation. Now, it has been widely recognized that they provide a useful framework for formulating many problems in mathematical finance. They are also useful for problems in stochastic control, stochastic differential game, and probabilistic formula for the solutions of quasi-linear partial differential equations. Thus he introduceda nonlinear mathematical expectation -g-expectation, which can define conditional expectation, according to the solution of BSDE:ε_0,T[ξ] = y₀.As a kind of nonlinear mathematical expectation, g-expectation has a lot of different properties from classical mathematical expectation. We list them without proving: constancy, monotony will be included. In the first chapter, we introduce not only BSDE and its comparison theorem, but also the simple properties of the risk measure.One of the main results in this paper is a necessary and sufficient conditionon the constancy of g-expectations which will be given in the second chapter. We have already known that:The following conclusion will also be obtained easily:Naturally, the issue we would like to explore is whether or not g(y, 0, t) = 0 whenε_0,T[α] =α? The answer is no. Example 2.4 and theorem 2.5 give the corresponding conclusions. We give two proven methods to theorem 2.5, one is based on the knowledge from mathematics analysis, the other is based on the comparison theorem of BSDE. So that we get the relationship between the constancy of g-expectations and the constancy of condition g-expectations. Furthermore, the relationship between the constancy of risk measure and the constancy of condition risk measure will be given. Finally, we prove through a theorem that, as a nonlinear operator, g-expectation ont only depends on g, but also on T.We check out the relationship between g-expectation and wealth process in chapter 3. At first we give a model of investment strategy process by a simple example of BSDE. And then we give two methods to solve special BSDE: the first, g = -[a_ty_t + b_tz_t] which is linear, the second, g(y_t,z_t,t) = a_t|z_t| which is simply nonlinear. We solve the first example by two methods, one is based on Girsanov transform, and the other is based on its dual equation.At last, in the fourth chapter, we do the work of their predecessors a summary and list the relationship among g, g-expectation and risk measure.And we also generalize converse comparison theorem of BSDE: from two parts to finite parts. We prove it by stopping time. Through checking the relationship between g-expectation and risk measure, we fasten the relationshipbetween the space of probability and the space of risk measure further more.