非线性期望理论下的极限定理及其金融风险度量中应用的研究

Based on some problem about coherent measures of risk and convex measures of risk in financial market, this project will study some limits theorems under the nonlinear expectation theory and applications to financial risk measures. Our research work mainly includes the following detailed contents: (1) establish some relationship between G-expectation and coherent measures of risk, convex measures of risk; (2) under the Choquet expectation frame, to study some laws of large numbers , theorem of large deviations and some problems related to risk measures; (3) applications to some other problems about financial risk measure in the nonlinear expectation expectation theory frame. By employing the theories of BSDE, nonlinear stochastic analysis and full nonlinear PDE, we establish some relations among some kinds of specific nonlinear expectation(for example, upper expectation. Choquet expectation, one-side moment of coherent measure of risk) and some limits theorems. We will further study some problems of the capital requirement in order to make the risk acceptable (i.e. pricing). We will also give the difference between nonlinear expectation and classical expectation in the problems of the capital requirement.

本项目基于金融风险度量中的相容风险度量和凸风险度量问题，开展在非线性期望理论框架下极限定理以及金融风险度量中应用等问题的研究。主要研究的主题包括：（1）刻画相容风险度量和凸风险度量与G-期望之间关系的问题；（2）Choquet期望意义下的大数定律和大偏差定理以及与之相关的一些金融度量问题；（3）在非线性期望下的大数定律的应用问题。本项目拟应用倒向随机微分方程、非线性随机分析和完全非线性二阶偏微分方程等相关理论建立以上研究主题的G-期望理论的有关内容，系统讨论和刻画各种不同的具体的非线性期望（如上期望、Choquet期望、单边矩相容风险测度等）之间的关系，以及其极限理论。基于这些理论我们进一步研究为避免风险所要的资本需求即定价等问题，并给出与传统资本需求（定价）问题之间的区别。

本项目基于金融风险度量中的相容风险度量问题，开展在非线性期望理论框架下极限定理以及金融市场中应用等问题的研究。主要研究了Choquet期望下的弱大数定律、强大数定律，提出了共单调随机集的概念；在次线性期望框架下，得到了与经典弱大数定律相一致的充分条件并在没有独立条件仅假设部分和矩条件情形下得到了强大数定律；在g-期望框架下通过与无穷区间的自由边界问题的联系研究了g-期望情形下最优停时的验证定理；另外在具有记忆效应影响的金融市场中，我们通过建立随机最大值原理得到了最优投资组合。