【作者】 王志攀；

【导师】 方世祖；

【作者基本信息】 广西大学 ， 应用数学， 2008， 硕士

【摘要】 在通常的风险模型中,往往假定保险公司中不同时期的保费收入和理赔额分别为两列独立同分布的随机变量,而且是相互独立的。但是,在保险公司实际经营中,索赔到达计数过程与保单到达计数过程是相依的,且险种呈现多元化,有必要为这类险种情形提供更为客观实际的风险模型,另外,自国际著名学者Hans U.Gerber和Elias S.W.Shiu于上世纪末首次提出破产时刻折现罚金函数的概念,风险理论中的一些有兴趣的重要精算量都是破产时刻折现罚金函数的特例。破产时刻折现罚金函数作为一个有力的数学工具,使得可以用一种统一的方式分析破产时刻、破产前瞬间盈余、破产时赤字以及相关的精算量。本论文建立并研究了三类风险模型:(一)考虑了保费率随机、保费收取过程是Poisson过程,而索赔计数过程是其稀疏过程的双险种风险模型的不破产概率问题,求出了不破产概率满足的积分方程,并在指数分布的情况下求出了无限时间不破产概率的具体表达式。(二)研究了保费率随机、保费收取过程是Poisson过程,而索赔计数过程是其稀疏过程的带干扰的双险种风险模型,讨论了其盈余过程的基本性质,强马氏性和鞅性,利用鞅证明了Lundberg不等式和最终破产概率的一般公式。(三)考虑了对于给定的初始状态和初始分布,保费率受马氏过程控制的风险模型,利用向后差分法得到了折现罚金函数以及条件破产概率所满足的积分方程,并推出了在具有平稳初始分布时折现罚金函数的递归不等式和零初始资产时破产概率的一个简洁估计。更多还原

【Abstract】 In actuarial science,the usual models are usually based on the indepeneny assumption,that is,the premiums and claims are assumed to be two independent and identically distributed(iid) randon variables series, and diferent times of policies are independent of each other. But in the insurance company physically the management, counting process claims and premiums arrived at the counting process is dependent, and grows and presents a diversification near there is necessity to for this type of grow situation and provide more objective and actual risk model nearly. Addition, at the end of last century, the conception of the expected discounted penalty function at ruin was first introduced by Hans U. Gerber and Elias S. W. Shiu who are contemporary international leading experts at ruin theory. A number of particular cases of the expected discounted penalty function at ruin led to important quantities of interest in risk theory. The expected discounted penalty function at ruin being a powerful analytical tool made it possible to analyze the time of ruin, the surplus immediately before ruin, the deficit at ruin, and related quantities in a unified manner. In this thesis, we build up and study three classes of risk model:(1) We consider the survival probability problem of a double-type risk model in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is thinning process. A integral equation for the survival probability are gotten. The explicit formula of the survival probability for the infinite interval is obtained in the special case-exponential distribution.(2) We study the ruin probability problem of the double-type risk model perturbed in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is thinning process. Using martingale method, the Lundberg inequality and the common formula for the ruin probability are proved.(3) To be consider for the initial state and initial distribution, premium rates markov process control by the risk model, by a backward differential argument, the integral equation satisfied by the expected discounted further more we obtain a recursive inequality about the expected discounted penalty with the stationary initial distribution and a simplified estimation of ruin probability with no initial reserve.更多还原