【作者】 王颖；

【导师】 刘国欣；

【作者基本信息】 中南大学 ， 概率论与数理统计， 2006， 博士

【摘要】 周知，经典风险模型是一个时齐的具有平稳独立增量的随机过程，对经典风险模型的研究已基本完善，各种保险精算量都得到了完整精确的分析表达式。本论文致力于对几种不同的风险模型进行研究，它们包括：连续时间复合二项模型，马氏环境下的连续时间复合二项模型，复合二项—Poisson模型以及带干扰的经典风险模型。主要研究了连续时间复合二项模型破产概率的一些性质及其期望折扣罚函数；马氏环境下的连续时间复合二项模型与复合二项—Poisson模型破产概率的上、下界限和逼近；带干扰的经典风险模型破产严重性的一些度量。 文章由六部分组成：(1)导论；(2)连续时间复合二项模型；(3)连续时间复合二项模型的期望折扣罚函数；(4)马氏环境下的连续时间复合二项模型；(5)复合二项—Poisson模型；(6)带干扰的经典风险模型破产严重性的度量。 导论部分主要介绍了风险理论方面的一些实际背景和数学描述，论题的历史和现状以及论文的内容提要。导论之后是论文的主体部分，包括以下内容： 我们首先建立了连续时间复合二项模型。连续时间复合二项模型可以看作是复合二项模型的时间连续化，其骨架链与复合二项模型一致，而其极限情形便是经典风险模型。据我们所知，这个模型首次将相邻两次索赔到达之间的时间间隔为离散型的随机变量纳入到时间连续的模型中研究。在连续时间复合二项模型中，相邻两次索赔到达之间的时间间隔服从离散无后记忆分布—几何分布，因此该模型的盈余过程是马氏过程，进而也是逐段决定马氏过程(PDMP)。在本论文的第二章我们将该模型纳入到PDMP框架中构造出鞅，借助测度变换和更新定理等方法，首次得到了连续时间复合二项模型破产概率的Lundberg界、Cramer-Lundberg逼近、破产概率的一般表达式以及当初始资金为0时破产前瞬间盈余与破产赤字的联合分布和破产概率的精确表达式。 在本论文的第三章主要研究连续时间复合二项模型的期望折扣罚函数。Gerber和Shiu(1998)将风险模型中三个重要的精算量：破产时间、破产前瞬间盈余和破产赤字嵌入到期望折扣罚函数中研更多还原

【Abstract】 It is well known that the classical risk model is an important stochastic process with properties of temporal homogeneity and independent increment. The study on this model is nearly perfect and exact calculated results for all actuarial diagnostics are derived in analytical form. In this dissertation, we consider several different risk models, including continuous-time compound binomial risk model, continuous-time compound binomial risk model in a Markovian environment, compound binomial-Poisson model and classical risk model that is perturbed by diffusion. We mainly discuss some properties of ruin probability and expected discounted penalty function for the continuous-time compound binomial risk model; Lundberg bounds and Cramer-Lundberg approximations for the continuous-time compound binomial risk model in a Markovian environment and the compound binomial-Poisson model; some measures of the severity of ruin in the classical risk model.The dissertation is consist of six parts: (1) Introduction; (2) Continuous-time compound binomial risk model; (3) Expected discounted penalty function for the continuous-time compound binomial risk model; (4) Continuous-time compound binomial risk model in a Markovian environment; (5) Compound binomial-Poisson model; (6) Some measures of the serverity of ruin in the classical risk model.A brief review of the background of risk theory and the relational history of the thesis as well as the main contents of the dissertation are given in the introduction. After that the main body of the dissertation starts.At first, we construct continuous-time compound binomial risk model. Continuous-time compound binomial risk model is a continuous-time version of the compound binomial risk model in discrete time, its skeleton chain is coincided with the compound binomial model and its limiting case is compound Poisson model. As we know, it is the first model that consider the discrete inter-occurance time random更多还原