非线性期望的极限性质及其在金融风险中的应用

This project is devoted to the enrichment and development of the theories of nonlinear expectation and backward stochastic differential equation, and investigates their applications in financial risks. It belongs to the cross-over study of nonlinear expectation, backward stochastic differential equation and mathematical finance, which are discussed by using the methods of nonlinear stochastic analysis, partial differential equation, probability limit theory, stochastic process and financial statistics. In this project, the following four aspects are considered. First, we discuss limit properties of a sequence of independent random variables without the requirement of identical distribution and a sequence of random variables satisfying more general conditions under sublinear expectations. Second, under convex expectations,limit properties of a sequence of random variables are considered. Third, limit properties of G-Brownian motion paths are investigated. At last, we discuss limit properties of a sequence of random variables under g-expectations induced by backward stochastic differential equations, generators of which satisfy Lipschitz conditions or more general conditions. These problems not only have application backgrounds in financial institutions, but also for nonlinear expectations and backward stochastic differential equations, they are frontier issues with great difficulties and challenges.

本项目致力于丰富和发展非线性期望和倒向随机微分方程理论以及探讨它们在金融风险方面的应用，属于非线性期望、倒向随机微分方程与数理金融三个方面的交叉研究。在非线性期望和倒向随机微分方程的理论研究中，我们将运用非线性随机分析、偏微分方程、概率极限、随机过程以及金融统计理论中的方法和技巧来进行探讨。该项目主要从以下四个方面进行研究：一是研究在次线性期望下独立但不同分布甚至更一般条件的随机变量序列的极限性质；二是探讨凸期望下随机变量序列的极限性质；三是讨论G-布朗运动轨道的极限性质；四是考虑由倒向随机微分方程诱导产生的g-期望下随机变量序列的极限性质，其中倒向随机微分方程的生成元g满足Lipschitz甚至更一般条件。这些问题不仅在金融机构中有应用背景，而且它们对非线性期望和倒向随机微分方程理论本身而言，也是具有很大难度和挑战的前沿问题。

本项目致力于丰富和发展非线性期望和倒向随机微分方程理论以及探讨它们在金融风险方面的应用，属于非线性期望、倒向随机微分方程与数理金融三个方面的交叉研究。该项目主要从以下三个方面进行研究：一是研究在次线性期望下独立但不同分布甚至更一般条件的随机变量序列的极限性质，如大数定律，重对数律；二是讨论G-布朗运动轨道的极限性质，如G-布朗运动的增量到底有多大，G-布朗运动的连续模定理；三是探讨被次线性期望控制的凸期望下独立但不同分布随机变量序列的极限性质，如大数定律，中心极限定理。此外，关于倒向随机微分方程的解及其相关问题的一些结果也被得到。这些问题不仅在金融机构中有应用背景，而且它们对非线性期望和倒向随机微分方程理论本身而言，也是具有很大难度和挑战的前沿问题。