带有保险风险与金融风险的相依离散时和连续时风险模型中破产概率的渐近估计研究

In risk theory, the risk model with insurance risks and financial risks is one of the important risk models to investigate. Comparing with the classical renewal risk model, our model contains not only insurance risks considered by many researchers, but also financial risks, which means that an insrance company is allowed to invest its wealth into some risky and risk-free markets, so is more reasonable. In recent ten years, such an independent model has been systematically investigated, and some asymptotic results for ruin probabilities have been derived. This project will generalize the model by allowing some dependence structures, to obtain some further results by means of heavy-tail analysis. Firstly, by using the analysis for tail probability of the randomly weighted sum and the methods in the extreme value theory, we aim to investigate the asymptotic behavior of ruin probabilities in the discrete-time insurance risk model with two kinds of dependent insurance risks and financial risks. Secondly, we consider the continuous-time risk model with financial risks described by a geometric Levy (or Cadlag) process. We will study the asymptotics for ruin probabilities by utlizing some methods in risk theory and some properties of Levy (or Cadlag) processes. Finally, we aim to develop some algorithm based on Monte Carlo in order to get the simulation of ruin probabilities, and conduct some empirical analysis of ruin probabilities by some data from some insurance companies. This project will provide decision support for the risk maganagement of insurance companies.

带有保险与金融风险的风险模型是风险理论领域研究的热点模型之一。相对于已经相当成熟的更新风险模型，本模型不但考虑了传统的保险风险，也引入了金融风险，即保险公司可以将其资本投入到有风险和无风险的投资市场，这样的模型更加合理而实际。近十年来，研究者对独立模型进行了系统的研究，获得了破产概率的渐近结果。本项目将更进一步研究该风险模型，同时允许模型存在某些相依结构，从重尾分析的角度，将原有研究成果向前推进一步。首先，利用随机加权和的尾渐近分析和极值理论中的方法，研究带有两类相依保险风险与金融风险的离散时模型中破产概率的渐近性；其次，研究连续时模型，借助几何Levy甚至Cadlag过程刻画金融风险，通过对这些随机过程的理论研究及风险理论中的方法估计破产概率；最后，通过改进的Monte Carlo模拟算法，计算破产概率的近似值，并利用重大赔付数据进行实证分析。本项目研究将为保险公司风险管理提供决策支持。

本项目在各种相依结构下，对带有金融风险和保险风险的离散时和连续时保险风险模型中多种风险量的渐近问题进行了深入的探索，并利用数值模拟的方法模拟估计风险量以及研究其渐近效率。首先研究了基于相依随机变量序列极限性态的理论，包括相依随机变量部分和的尾渐近公式、多元非平稳高斯序列最大值和部分和的联合渐近分布、分数布朗运动的增量和标准化增量最大值的精确尾渐近性、几类随机偏微分方程的弱渐近性，以及二元区间树步长的极限分布等；其次，利用随机加权和及随机递推方程的方法，研究了两类带有保险与金融风险的相依离散时风险模型，其一是保险风险相依，而金融风险任意相依，其二是每一对保险风险与金融风险相依，得到了有限时、无限时和随机时破产概率的渐近估计公式，以及最大索赔额再保险协议下，累积损失的在险价值量和条件尾期望渐近估计；最后研究了一维和二维连续时模型，通过几何Levy过程来描述收益过程，分析获得了有限时和无限时破产概率的一致渐近估计。另外，还研究了带有变化保费收入率的扰动复合Poisson风险模型，借助于Laplace变换的方法，研究了Gerber-Shiu方程，以及破产时刻的Laplace变换，以及破产时刻赤字的估计等；同时，我们还利用了粗略蒙特卡洛方法，实现了对离散时和连续时模型中有限时破产概率及其其它风险量的统计模拟计算。