随机保费下扰动风险模型总索赔盈余的大偏差

2018-04-16 00:55董英华
经济数学 2018年1期

董英华

摘 要 考虑随机保费下带干扰的风险模型,其中保费额和索赔额各自形成了END的随机变量序列,保费次数是由一个拟更新过程描绘,干扰项是由一个布朗运动过程来刻画.在索赔额分布属于一致变化类的条件下,给出了总索赔盈余过程的精致大偏差.

关键词 应用概率论;精致大偏差;拟更新过程;END随机变量;总索赔盈余

中图分类号 O211.9  文献标识码 A

Abstract Consider a perturbed risk renewal model with investment income, in which premium sizes and cliam sizes form two sequences of END random variables, respectively, the premium number is described by a quasi-renewal process, and  the investment income is characterized by Brownian motion. Provided that claim-size distribution belongs to the consistent variation class, the precise large deviation for the surplus process of aggregate claims was presented.

Key words applied probability theory; precise large deviation; quasi-renewal process; END random variables; aggregate claim surplus

1 引 言

最近一些年來,研究保险风险模型中一些量的精致大偏差成为一大热点.例如,在索赔额相互独立的条件下,Ng等(2004) [1]研究了具有一致变化尾索赔额总量的精致大偏差.高珊和孙道德(2007)[2]考虑了双险种Poisson风险模型,当各险种索赔额相互独立并且具有广义正则变化尾时,给出了总索赔量的大偏差.另外,在现实生活中,保险公司的索赔额往往具有一定的相依性.因此,一些科研工作者考虑具有相依性随机变量和的精致大偏差问题.

例如,Tang (2006)[3]考虑了ND随机变量的非随机和的精致大偏差,而Liu (2009)[4], Chen等 (2011)[5]以及Wang等 (2013) [6] 对上述文献进行了推广,研究了END随机变量和的精致大偏差.韦晓等(2007) [7]考虑了变保费率带干扰的复合Poisson风险模型,干扰项是由一个有界的布朗运动所刻画,在索赔额分布属于广义正则变化族的条件下给出了总索赔盈余过程的精致大偏差.而Chen等(2014) [8] 研究了广义复合更新风险模型,其中索赔额形成了ND的随机变量序列,在索赔额分布具有一致变化尾的条件下,给出了总索赔盈余过程的精致大偏差.本文在文献[7]和[8]的基础上,考虑更一般的风险模型,其中索赔额和保费额各自形成了END的随机变量序列,保费次数是由一个拟更新过程描绘,干扰是由一个无界的布朗运动过程所刻画.在索赔额分布属于一致变化类的条件下,给出了总索赔盈余过程的精致大偏差.

2 END的定义和模型的描述

在这节,假定保费额和索赔额都是END的.为此,首先给出END随机变量的定义.

6 结 论

本文研究了保费收入为拟更新过程的带干扰风险模型.在索赔额具有一致变化尾的条件下,讨论了总索赔量过程和总索赔盈余过程的大偏差.结果显示出当索赔额具有一致变化尾时,总索赔量过程精致大偏差的渐近行为只与索赔数的更新函数与索赔额的分布有关,而与投资收入扩散过程无关;另外,总索赔额盈余过程精致大偏差的渐近公式与总索赔量精致大偏差的渐近公式是相同的,这说明总索赔盈余过程的精致大偏差的渐近行为与随机保费收入过程无关.

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