风险理论中的渐近分析及其应用

This project will be devoted to asymptotic analysis of risk processes and their applications in insurance and finance focusing on two main research directions: 1) asymptotics in extended renewal risk models with various dependence structures, and 2) the interplay between insurance and financial risks in the sense of asymptotics. . Our motivation for the first topic stems from the major drawback of the classical renewal risk model related to the independence assumptions between both claim sizes of an insurance portfolio and the claim inter-arrival times. Despite its mathematical tractability the classical renewal risk model is far away from practical situations. Therefore, in this project we propose to deeply study extended renewal risk models with various dependence structures which are appealing for practical applications.. When allowing for dependence in risk models the conventional methods and techniques will fail. Therefore, our main approach to tackle these issues will be in spirit of asymptotics. In order to study the asymptotic behaviour of insurance quantities under consideration, the methodological base will be the extreme value theory (EVT). Recent progress of EVT opens the way for dealing with complex multivariate distribution structures. Based on existing literatures, we plan to propose some new types of extended renewal risk models with tractable dependence structures, and then derive asymptotic formulas for various important insurance quantities, including ruin probabilities and the tails of aggregated risks.. The second main direction of this project investigates the interplay between insurance and financial risks in the sense of asymptotics. A natural question in risk theory is which one of the insurance risk and financial risk plays a dominating role on the ruin probability of an insurance company. This question concerns the interplay between the two types of risk and unavoidably leads to a complicated stochastic structure for the wealth process of the insurance company. . In order to study this direction we will consider jump-diffusion risk models such as the bivariate Lévy-driven risk model. The techniques and methods of modern stochastic processes and stochastic analysis that will be employed here include martingale methods, stopping-time techniques, and stochastic control theory. In fact, we have considered this topic for a discrete-time risk model and achieved some remarkable results. Based on our obtained results, we shall extend the investigation to continuous-time risk models which will lead some ground-breaking results for the bivariate Lévy-driven risk model.

本项目将致力于风险过程的渐近分析及其在金融保险中的应用，其中两个主要研究方向分别是1）带有各种相依机构的扩展风险模型的渐近性质以及2）渐近意义下金融风险和保险风险之间的相互作用关系。当允许风险模型中存在相依结构时，传统基于独立性的方法和技巧将失效。因此在第一个研究课题中我们将利用渐近的思想处理遇到的问题，所用的主要方法来源于极值理论。我们力争在标准更新风险模型中引入一些具有可操作性的相依结构，并在新模型下推导出关于破产概率、累计风险尾概率等重要保险量的渐近公式。对于第二个课题的研究将涉及一些复杂的跳扩散风险模型，比如由二元Lévy过程驱动的风险模型。我们将使用近现代随机过程和随机分析的经典方法和技巧，包括鞅方法、停时技巧、以及随机控制理论等。事实上，我们已就一类离散时间风险模型展开了此方向的研究并取得了一些很好的成果，在此基础上本项目将力争把已得结果推广延伸到更复杂的连续时间风险模型。

本项目中我们深入研究了风险过程的渐近性质及其在金融保险中的应用，其中两个主要的研究方向分别是1）带有各种相依机构的扩展风险模型的渐近性质以及2）渐近意义下金融风险和保险风险之间的相互作用关系。当允许风险模型中存在相依结构时，传统基于独立性的方法和技巧将失效。因此在第一个研究课题中我们利用了渐近的思想处理遇到的问题，所用的主要方法和技巧来源于极值理论。我们在标准更新风险模型中引入一些具有可操作性的相依结构，并在新模型下推导出关于破产概率、累计风险尾概率等重要保险量的渐近公式。对于第二个课题的研究涉及一些复杂的跳扩散风险模型，比如由二元Lévy过程驱动的风险模型。本项目中我们就一类离散时间风险模型展开了此方向的研究并取得了一些很好的成果，在此基础上未来我们将力争把已得结果推广延伸到更复杂的连续时间风险模型。本项目执行期间，我们共发表SCI索引论文9篇，参加国际会议并做分组报告4次。