含有投资回报和相依结构的保险风险模型破产概率的渐近性研究

According to the tail asymptotics of randomly weighted sums, the project will study the asymptotic behaviors of ruin probabilities in the insurance risk models with dependence structures and return on investments. Firstly, we will discuss the asymptotic behaviors of the tail probabilities of the randomly weighted sums and their maximum with heavy-tailed random variables satisfying a certain dependence structure, and then investigate the asymptotics of ruin probabilities in one-dimensional and multi-dimensional discrete-time risk models with dependent insurance risks and dependent financial risks; meanwhile, we also consider the case when the insurance risks and financial risks are mutually dependent. Secondly, we will study the uniformly asymptotic estimate for the finite-time (absolute) ruin probabilities and precise large deviations in some types of continuous-time heavy-tailed risk models with constant rate of return on investments (i.e. constant force of interest) and dependence structures, where these models we consider are, for example, jump-diffusion model, multi--dimensional risk model, Cox model, delayed-claim risk model and others. Thirdly, we will deal with the models mentioned above with random return on investments (i.e. stochastic interest), where we also discuss the optimal constant investment strategy. Finally, we will give the statistics implementation for ruin probabilities by the statistical simulation and its corresponding algorithm. Hence, the project is of important academic significance in theory and potential value in application so as to provide theoretical foundations on the risk evaluation and the scientific decision for insurance company.

本项目拟借助于随机加权和的尾渐近性理论，研究含有投资回报和相依结构的几类保险风险模型破产概率的渐近性质，主要体现在以下四个方面：.一、研究满足某种相依结构的随机加权和及其最大值的尾渐近性，进而讨论兼有相依保险风险和相依金融风险的一维和多维离散时间风险模型破产概率的渐近性，同时亦考虑保险风险与金融风险之间也具有某种相依结构的情形．.二、研究含有固定投资回报率和相依结构的几类连续时间风险模型（如跳-扩散模型、多维风险模型、复合风险模型、Cox模型及延迟索赔风险模型等）的有限时（绝对）破产概率的一致渐近性，以及总索赔额的精致大偏差等．.三、研究上述连续时间风险模型在随机投资回报下的破产概率的渐近性，同时亦探讨最优常数投资策略等．.四、利用统计模拟及相应算法对破产概率的理论结果加以统计实现．.本项目的研究具有重要的理论意义和潜在的应用价值，为保险公司进行风险评估和科学决策等提供理论依据和技术支持．

周知，破产理论是保险风险理论的核心内容，也是当前应用概率统计的重要分支，是对金融保险风险进行定量分析和预测的基础理论。根据项目计划书，本项目主要借助于随机加权和的尾渐近性理论，分别考虑了含有投资回报和相依结构的若干保险风险模型，系统地研究了有限时和无限时破产概率、索赔总额的尾概率和精致大偏差等的渐近性或一致渐近性，完成了项目计划书提出的研究目标和研究内容，所得到的结果对已有的相应成果做了实质性的推广和改进。本项目主要的难点和创新点是在含有相依结构、投资回报和重尾索赔场合下讨论保险风险模型的渐近行为。.本项目的重要结果主要体现在以下四个方面：.一、研究了满足广义条件相依结构的无穷加权和、随机加权和及其最大值的尾概率的渐近性或一致渐近性，进而讨论了兼有相依保险风险和相依金融风险的离散时间风险模型破产概率的渐近性或一致渐近性；.二、研究了含有常利息力和相依结构的几类连续时间风险模型，如二维风险模型、一重或多重延迟索赔风险模型等；当索赔额分布属于某一重尾分布族时，讨论了有限时破产概率的一致渐近行为和无限时破产概率的渐近行为；.三、研究了含有常利息力和相依结构的非标准更新风险模型，考虑了当索赔额分布属于某一重尾分布族时的索赔总额折现值尾概率的一致渐近性质；.四、研究了含有利率或无利率的非标准(复合)更新风险模型，当模型满足某种相依结构以及索赔额分布属于某一重尾分布族时，得到了索赔总额或其折现值的精致大偏差的渐近性。.本项目属基础性研究，主要成果为学术论文。现已完成11篇高水平论文，其中7篇论文在SCI期刊发表或被录用待发表，达到了项目计划书的研究目标。本项目研究具有重要的理论意义和潜在的应用前景，为保险公司进行风险评估和科学决策等提供理论依据和技术支持。所得成果预计在5-10年或更长时期内推广使用。另外，本项目亦讨论了应用概率的其他有意义的问题，等等。