倒向随机微分方程和非线性期望几个新问题的研究

This project is devoted to the enrichment and development of the theories of backward stochastic differential equation and nonlinear expectation. It belongs to the cross-over study of backward stochastic differential equation, nonlinear expectation and financial mathematics, which are discussed by using the methods of nonlinear stochastic analysis and stochastic process. In this project, the following three aspects are considered. First, under non-Lipschitz condition, we investigate the existence and uniqueness of the solution of backward doubly stochastic differential equation and the related problems such as, comparison theorem, converse comparison theorem. And the probabilistic representation of nonlinear stochastic partial differential equation is given by using the solution of backward doubly stochastic differential equation. Second, under non-Lipschitz conditions, the existence and uniqueness results of the solutions of anticipated backward doubly stochastic differential equation and anticipated backward doubly stochastic differential equation with reflection, and the related problems such as, comparison theorems and converse comparison theorems are considered. And we get some results on the problems of the above equations in stochastic control. At last, in the framework of G-expectation, we discuss some theorems such as Fubini theorem, Lusin’s theorem and Egoroff’s theorem. These problems not only have application backgrounds in financial mathematics, but also for backward stochastic differential equations and nonlinear expectations, they are frontier issues with some difficulties and challenges.

本项目致力于丰富和发展倒向随机微分方程和非线性期望理论，属于倒向随机微分方程、非线性期望与金融数学三个方面的交叉研究。在倒向随机微分方程和非线性期望的理论研究中，我们将运用非线性随机分析和随机过程理论中的方法和技巧来进行探讨。该项目主要从以下三个方面进行研究：一是研究生成元满足非Lipschitz条件下，倒向重随机微分方程解的存在性、惟一性以及比较定理、逆比较定理，并通过其解给出非线性随机偏微分方程的概率解释；二是考虑生成元满足非Lipschitz条件下，超前倒向重随机微分方程和超前反射倒向重随机微分方程解的存在性、惟一性以及比较定理、逆比较定理，并得到以上两类方程在随机控制问题中的一些结果；三是探讨G-期望框架下的Fubini定理、Lusin定理、Egoroff定理。这些问题不仅在金融数学中有应用背景，而且它们对倒向随机微分方程和非线性期望理论本身而言，也是具有一定难度和挑战的前沿问题。

本项目致力于丰富和发展倒向随机微分方程和非线性期望理论，属于倒向随机微分方程、非线性期望与金融数学三个方面的交叉研究。在倒向随机微分方程和非线性期望的理论研究中，我们将运用非线性随机分析和随机过程理论中的方法和技巧来进行探讨。该项目主要从以下三个方面进行研究：一是研究生成元满足非Lipschitz条件下，倒向重随机微分方程解的存在性、惟一性以及比较定理、逆比较定理，并通过其解给出非线性随机偏微分方程的概率解释；二是考虑生成元满足非Lipschitz条件下，超前倒向重随机微分方程解的存在性、惟一性以及比较定理、逆比较定理，并得到该方程在随机控制问题中的一些结果；三是探讨g-期望框架下的Fubini定理、Lusin定理、Egoroff定理。此外，关于非线性期望的一些结果也被得到。这些问题不仅在金融数学中有应用背景，而且它们对倒向随机微分方程和非线性期望理论本身而言，也是具有一定难度和挑战的前沿问题。