几类相依的双险种风险模型破产问题研究

【作者】 张子辉；

【导师】 俞政；

【作者基本信息】 中南大学 ， 概率论与数理统计， 2009， 硕士

【摘要】 自从经典的Cramer-Lundberg风险模型提出以来,许多人都对其进行了推广,但往往都蕴含着这样一种假定：保险公司中不同险种的索赔到达计数过程是相互独立的,即不同险种的理赔额是相互独立的；保费到达计数过程与索赔到达计数过程也是相互独立的,即是保费收入和理赔额是相互独立的随机变量。但是,在保险公司的实际经营中,由于竞争,利率,通货膨胀率以及随机干扰项等经济环境的影响,不同险种的索赔到达计数过程是相依的,保单到达计数过程与理赔到达计数过程也是相依的,根据这一实际情况,有必要为这类险种提供更符合客观实际的风险模型。本文建立并探讨了以下几种相依的双险种风险模型：(1)讨论了常利率和通货膨胀率下一类索赔到达过程相依同时保费收入为复合Poisson过程的双险种风险模型的破产概率,推算了调节系数,破产概率之间的关系等问题。先将两个相依索赔总额转化为相互独立的索赔总额,然后利用鞅方法给出相应的Lundberg不等式。(2)考虑了每次收取的保费均为独立同分布的随机变量,保费到达计数过程是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生存概率问题,求出了生存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。(5)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,并考虑了利率和通货膨胀率,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。更多还原

【Abstract】 The classical Cramer-Lundberg risk model has been promoted by many people since it was proposed. While they are based on the independent assumptions, these are, the counting process in different types of insurance claims is independent of each other; counting process claims and premiums arrived at the counting process are assumed to be independent. In other words, the claims of different kinds of insurance are independent and the premiums and claims are assumed to be two indepe-ndent and identically distributed(iid) random variables series, anddifferent times of polices are independent of each other. In the managemen of the insurance company, because of economic impact of competition, interestrate, inflation rate and random interferemces, the counting processes in different types of insurance claims are dependent, counting process claims and premiums arrived at the counting process are also dependent. There is necessary for this type of grow situation to provide more objective and actual risk model nearly. In this thesis we build up and discuss several kinds of dependent double-type risk models:(1) We study a correlated aggregate claims model,in which the arrival of the income of premium is a compound poisson process with constant interest and inflation rate.First we convert the two correlated aggregate claims to independent aggregate claims. Then we get Lundberg inequality by using martingle theory.(2) We consider the Probability properties 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 is obtained. The explicit expression of the survival probability for the infinite interval is obtained in the special case of exponential distribution.(3) 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.更多还原