扭曲风险度量及其尾渐近性的研究

Risk measures are the fundamental tools used in almost all topics of risk management which are extensively used in finance and insurance. This project aims to study distortion risk measure and its tail asymptotics. We use three methods to construct new class of distortion functions and measures. The first one is composting methods, the second one is mixing methods and the third approach is based on the theory of copula. Subadditivity is an appealing property when aggregating risks in order to preserve the benefits of diversification. However, risk measures are not always globally subadditive, such as value at risk which is not globally subadditive except of elliptically distributed risks. In this project, we use the latest limit theory of heavy-tailed distributions, the extreme value distribution theory and the theory of copula to investigate the tail sub and superadditivity instead of sub and superadditivity for value at risk and other distortion risk measures under various dependence and heavy-tailedness for risks. More generally, we will study the tail asymptotic behaviour for distortion risk measures, some sufficient and necessary conditions are derived. Various examples are also presented to illustrate the results. Our new foundings not only enrich the theory of risk management and insurance mathematics but also provide the theoretical foundations on the risk evaluation of the business and supervision for banking and insurance company.

扭曲风险度量是风险管理的基本工具,它广泛应用于金融、保险等领域。本项目旨在构建新的扭曲度量，研究其尾部渐近性。我们拟使用三种方法构建新的扭曲函数和扭曲度量。第一是复合方法，第二是混合方法，第三种方法是基于Copula理论。当我们研究聚集风险的多样化的影响时，次可加性是一个有吸引力的性质。然而，扭曲风险度量并不总是（全局）次可加的，如在险价值（除椭圆分布的风险外）就不是全局次可加的。在本项目中，我们利用重尾分布的极限理论、极值分布理论、copula理论及保险风险模型的最新成果， 在各种相依关系及重尾条件下，研究风险度量的尾部可加性、尾部次可加性与尾部超可加性。更一般地，我们将研究扭曲风险度量的尾部渐近性，推导出充分必要条件，并给出一些应用的例子加以说明。通过本项目的实施，所获得的研究结果不仅丰富和发展风险管理与保险数学的内容，而且对企业的风险评估、银行和保险公司的监管也提供理论基础

本项目旨在构建新的扭曲度量，研究其尾部渐近性并给出在风险管理等方面的应用。用三种方法构建了新的扭曲函数和扭曲度量。第一是复合方法，第二是混合方法，第三种方法是基于Copula理论。当我们研究聚集风险的多样化的影响时，次可加性是一个关键的性质。然而，扭曲风险度量并不总是（全局）次可加的。在本项目中，我们利用重尾分布的极限理论、极值分布理论、copula理论及保险风险模型的最新成果， 在各种相依关系及重尾条件下，研究了风险度量的尾部可加性、尾部次可加性与尾部超可加性。进一步，我们研究了扭曲风险度量的尾部渐近性，推导出充分必要条件。 在支撑多属性决策者的选择决策理论、资产配置、再保策略、最优互惠再保策略、有限制和无限制的最优再保、随机比较等方面给出了应用。