Non-Lipschitz Backward Stochastic Differential Equations And G-Expectations

The present thesis mainly consists of two parts.In the first section, we discuss a sort of backward stochastic differential equations and the properties of its solution udder the conditions that are given by Mao Xuerong. This section is mostly benefited from Peng's results.Firstly, making use of the notion of g-supersolutions, we get the limit theory of solutions of BSDE, and furthermore taking advantage of the limit theorem, we get nonlinear Doob-Meyer decomposition of g-martingales under Mao's conditions.Secondly, some peculiar conditions are attached on the drift coefficients. Under these conditions, we introduce the notions of g-expectations, conditional g-expectations, and discuss their properties that are similar to the classical situations except linearity. We also discuss g-martingales simply and their continuance, bounded stopping theorem, and upcrossing inequality.Finally, favored by the article "Stochastic differential utility" written by Duffle and Epstein, we define the notion of Stochastic differential utility under Mao's conditions. Similar to common utility functions, they preserve many good characters such as continuance, monotonousness, risk aversion and so on.In the second section, in infinite horizon, we discuss the solution of BSDE. This section is mainly benefited from Chen Zengjing's thesis in the degree of doctor.Firstly, g-expectations are usually defines on the space of square integrable random variables. Here we extend the domain to the space of integrable random variables. This is draw on the extension theory of operators. Secondly, in infinite horizon, we obtain the limit theory of g-supersolutions.