离散时间比例再保险模型的破产概率

2018-04-16 00:55王旭
经济数学 2018年1期

王旭

摘 要 研究具有相依結构的离散时间比例再保险模型的破产概率.在模型中假设随机利率和索赔间隔时间是相依的.利用更新递归技巧,首先得到了破产概率满足的递归方程.然后,根据该递归方程得到了破产概率的上下界估计.

关键词 破产概率;比例再保险;相依结构

中图分类号 O211.67  文献标识码 A

Abstract This paper studied  the ruin probability of discrete time proportional reinsurance model with dependent structure. Assuming that the stochastic interest rate depends upon the inter-arrival time, the recursive equations for ruin probabilities were derived by using the recursive renewal techniques. Then, the upper and lower bounds were obtained in terms of the recursive equation.

Key words bankruptcy probability; proportional reinsurance; dependence structure

1 引 言

保险公司的破产概率一直是风险控制理论的研究热点.在经典的风险模型中,盈余过程假定具有平稳独立增量性质.然而从保险业的现实角度出发,这种假设条件显然不切实际.因此,保险精算理论学者对经典风险模型进行了各种推广.对于离散时间模型,Cai[1,2]分别假设利率为一阶相依的自回归结构和Markov链,研究了破产概率满足的递归计算公式和上界估计;Xu[3]研究了一类具有Markov链利率和随机投资回报的离散时间风险过程的破产概率最小上界问题;牛祥秋[4]研究了具有Markov链利率的比例再保险模型,分别用递归更新方法和鞅方法得出了破产概率的上界; Dam和Chung[5]研究了成数再保险问题,得到了保险人和再保险人的联合破产概率的计算公式;而Diasparra和Romer[6]研究了具有马尔可夫利率链的离散时间风险过程的比例再保险模型的破产概率,得出破产概率满足的递归方程,给出了破产概率的广义伦德伯格不等式.

然而,上述文献中仅假设利率本身具有相依结构,而利率和索赔间隔时间以及索赔额相互独立,而现实中的利率一般是依赖于时间的.基于这个事实,本文考虑一个利率与索赔间隔时间相依的离散时间比例再保险模型,得到了破产概率满足的递归方程以及破产概率的上下界估计.众所周知,除一些特殊情况外,在理论上很难获得破产概率的解析表达式,一种行之有效的办法是给出破产概率的上下界.本文获得的结果可以为保险企业提供一定的决策参考.

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