具有延迟索赔的风险模型破产理论及相关问题研究

In the classical risk model, in case of the risky events occur, the claims for damage and compensation are triggered automatically and immediately . Whereas, in practical situation, the insurance claims may lag behind the events due to various reasons. It is necessary to study the ruin theory for the risk model allowing for delay in claims settlements. Basing on Gerber-Shiu discounted penalty function, we invstigate the ruin theories and related problems for the renewal risk model with delayed claims, the risk model with delayed claims and dividend payments and the risk model with dependence between claim sizes and the occurrences of delayed claims by martingale theory, Laplace transform, renewal argument and stochastic analysis. The expressions of ruin probability， the distribution function of the surplus before ruin，the distribution of surplus immediately before ruin and the deficit at ruin, the moments of the discounted sum of the dividend payments until ruin are obtained. The impact of the dependence between claim sizes and the occurrences of delayed claims on the ruin function is illustrated. The study on ruin theory for risk model allowing for delay in claims settlement in this project aims at enriching the content of ruin theory, as well as providing the theoretical analysis and technical support for insurance company.

在经典的风险模型中，假设风险一旦发生，索赔也立即发生。然而，在实际中，由于各种原因，索赔可能滞后于风险发生。这种现象客观存在于多种保险业务中，因此，迫切需要研究索赔延迟发生风险模型的破产理论。本项目以Gerber-Shiu折现罚金函数为研究工具，利用鞅、拉普拉斯变换、随机过程、更新方程等理论和方法，分别对具有延迟索赔的更新风险模型、具有延迟索赔和红利分配的风险模型、索赔延迟和索赔额相关的风险模型展开研究，探索这几类风险模型的破产理论及相关问题，得出破产概率等破产函数以及折现分红总量现值高阶矩的表达式，分析索赔延迟发生与索赔额之间的相关性对破产概率等破产函数的影响。本项目旨在推广经典风险模型破产理论的相应结果，为保险公司分析索赔可能延迟发生这一特性对公司盈余的影响提供理论基础和技术支持。

本课题主要对具有延迟索赔的风险模型的破产理论及相关问题展开了较为深入的研究。项目以Gerber-Shiu折现罚金函数为研究工具，利用拉普拉斯变换、随机过程、更新方程等理论和方法，分别对具有延迟索赔和红利分配的风险模型、索赔延迟和索赔额相关的风险模型、具有随机利率结构的延迟索赔风险模型展开了研究，计算出了Gerber-Shiu折现罚金函数的显示表达式，得出了破产概率的解析表达式，证明了Gerber-Shiu折现罚金函数和折现分红总量期望现值的关系式，并且分析了索赔延迟发生的概率、索赔延迟发生与索赔额之间的相关性对破产概率等破产函数的影响。研究成果以论文的形式发表在著名的精算学和应用概率统计杂志上，共发表论文（含接受3篇）12篇，其中10篇论文被SCI检索（含接受3篇SCI源刊论文）。本项目的研究工作推广了经典风险模型破产理论的相应结果，为保险公司分析索赔可能延迟发生这一特性对公司盈余的影响提供理论基础和技术支持。