风险模型的渐进与随机分析

Currently, as the economy is developing rapidly and human social activities are accelerating, various disaster risk events are occurring more and more frequently. We need to study appropriate statistical actuarial theory to establish risk models, observe the nature of their tail distributions and establish relevant risk prevention mechanisms. It has very important theoretical research and practical application significance...The research contents of this project are as follows: (1) Based on the extreme value theory tool, we intend to develop some new methods to observe risk measures, especially the methods to observe the robustness of extreme value risk measures, which are important for financial practice and theoretical exploration. (2) Analyze the occupation time of a geometric fractional Brownian motion-related asset hedging model. Geometric fractional Brownian motion is widely used as a typical risk asset model in the financial economy. Compared with Brownian motion, fractional Brownian motion is more and more considered because of its better tail behaviour and long-term dependence. (3) Study the liquidation probability of a Brownian motion risk model. In the research process of (2) and (3), the new extreme value theory methods and tools to analyze the tail distributions will be developed, which can be applied to more general theoretical research.

在经济快速发展，人类社会活动加速进行的当下，各种灾害风险事件越来越频繁发生。研究运用适当的统计精算理论方法建立适当的风险模型，观察其尾分布的性质并建立相关的风险预防机制，具有非常重要的理论研究以及实际应用意义。..在本项目研究内容有：（1）以极值理论工具为基础我们拟发展一些新的方法观察风险测度，特别是观察极值风险测度稳健性的方法，这对于金融实践以及理论探索都重要的意义。（2）分析一个几何分数布朗运动相关的资产对冲模型的占有时。几何分数布朗运动在金融经济当中作为典型的风险资产模型被广泛运用。与布朗运动相比，分数布朗运动因其具有更好的尾行为以及长期依赖性，也越来越多得被考虑。（3）研究一个布朗运动风险模型的清算概率。在（2）与（3）问题的研究过程中，新的极值理论尾分布的方法以及研究工具将会被发展，这些将能够被运用到更一般的理论研究当中。

本项目的研究内容涉及到风险管理、数理金融及保险精算等领域中的众多基础风险模型及风险测度，针对风险模型以及破产概率的理论结果及其广泛应用充分展示了其理论研究价值和实践指导意义。.研究成果包含如下四个方面：.1).对风险测度的研究当中，给出了一个重要的统计测度的极值分析估计结果。.2).对一个与利率以及税收有关的布朗运动风险模型的破产概率进行探讨和分析。.3).本研究将高斯极值理论推广的多维高斯场的情况，并将高斯极值理论运用到更一般和广泛的情况。.4).对连续极值在离散化空间上的情况进行了探讨和分析，给出了具有实际应用价值的分析结果。

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