【作者】 钱静静；

【导师】 王向荣；

【作者基本信息】 山东科技大学 ， 应用数学， 2004， 硕士

【摘要】 本硕士论文主要由两部分内容组成。 第一部分在毛学荣给出的条件下讨论一类倒向随机微分方程及其解的性质。这部分内容主要受益于彭实戈教授的相关结果。 首先借助于g—上解的概念，得到了此类倒向方程解的极限定理，并利用此极限定理证明了毛氏条件下的非线性Doob-Meyer分解定理。其次，对漂移系数附加特殊条件，讨论g-期望、条件g-期望以及它们所保持的类似于经典情况下的数学性质，并且简单讨论了g-鞅的连续性，有界停时定理以及上穿不等式。最后，受Duffie和Epstein的“Stochastic differential utility”一文的启发，在毛氏条件下定义了随机微分效用，它保持了一般效用函数的性质：单调性，连续性，风险厌恶以等。 第二部分在无穷水平上讨论倒向随机微分方程的解。这部分内容主要是受陈增敬教授的博士论文的启发。 首先，将定义在平方可积随机变量空间上的g-期望延拓到可积变量空间，这主要是用到算子的延拓定理。其次，在无穷水平上证明了g—上解的极限定理。更多还原

【Abstract】 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.更多还原