保费随机收入的二维风险模型的破产问题

【作者】 柳素秀；

【导师】 吕玉华；

【作者基本信息】 曲阜师范大学 ， 概率论与数理统计， 2010， 硕士

【摘要】 风险理论是保险数学中的重要理论,而破产论又是风险理论的核心.破产论最早提出的风险模型是Lundberg-Cramer经典破产模型,随着人们对这一模型全面而系统的研究,使经典论的研究内容不断得到深化.如,保费收入随机化的风险模型就是其中改进后得到的一种新模型.本文将保费随机收入的一维风险模型推广到二维风险模型,即为保费随机收入的二维风险模型.但是二维风险模型并不是一维风险模型的简单推广,它要比一维风险模型复杂得多.本文用两种方法探讨了保费随机收入的二维风险模型的破产问题.根据内容本文分为以下三章：第一章为绪论,主要介绍风险模型发展的历史、二维风险模型的研究现状和部分相关的概率知识.第二章讨论了保费随机收入的二维风险模型的破产问题.首先用鞅方法求出它的Lundberg型上界然后假定当保险公司收取的两种保费(C1j,C2j),J=1,2,…和支付的两种索赔(X1j,X2j),j=1,2,…均服从指数分布,并且它们的联合分布函数F1(c1j,c2j),j=1,2,…和F2(x1j,x2j),j=1,2,…均属于二元的Farlie-Gumbel-Morgenstern类,当关联系数p1和p2发生变化时,通过数据分析了破产概率上界的变化情况.第三章主要是用一维风险过程来讨论二维风险模型的破产问题.首先,定义一个一维风险模型其中参数(a1,a2)满足ai>0,i=1,2且∑i=12ai=1.这样,第二章中定义的二维风险模型转化成了一维模型的形式,其中从而,运用一维风险过程的结果得到了保费随机收入的二维风险模型最终破产概率满足的Lundberg不等式及最终破产概率满足的解析式然后,运用一维风险过程的方法推导出了它的Gerber-Shiu罚金折现函数Φ(ua)满足的积分方程并求出Φ(ua)在n=1时的Laplace变换之后验证了Φ(r)在特殊情况下与一维风险模型中的结果是一致的.最后探讨了保费随机收入的二维风险模型推广了的Gerber-Shiu罚金折现函数更多还原

【Abstract】 Risk theory is an important part in insurance mathematics, and ruin theory is the main content in risk theory. Lundberg-Cramer classical risk model is the first-emerging model in ruin theory. The contents of classical theory are deepened with people’s com-prehensive and systematic research. For example, the risk model with random premium income is one of improved models. In this paper, a one-dimensional risk model with random premium income is extended to a two-dimensional risk model which is a two-dimensional risk model with random premium income. Of course, the two-dimensional risk model is not a simple extension of the one-dimensional risk model and it is more complicated than a one-dimensional model. In this thesis, we discuss the ruin problem of the two-dimensional risk model with stochastic premium through two methods.This thesis is divided into three chapters according to contents:Chapter 1 is prolegomenon, in which we mainly introduce the developmental history of risk model, the current situation of a two-dimensional risk model’s research and part of interrelated knowledge of probability.In chapter 2, we discuss the ruin probability of a two-dimensional risk model with random premium income which is First, we obtain a Lundberg-type upper bound through the martingale technique. Then, by supposing that two kinds of premiums (C1j,C2j), j= 1,2,…and two kinds of claims (X1j,X2j), j= 1,2,…all obey ex-ponential distribution and their joint distribution functions F1(c1j,C2j),j= 1,2,…and F2(x1j,X2j),j= 1,2,…belong to bivariate Farlie-Gumbel-Morgenstern class, we analyse the upper bound’s change of infinite-time ruin probabality by analysis of data when the correlation coefficientρ1 andρ2 change.In chapter 3, we mainly consider the ruin problem of the two-dimensional risk model through methods of the one-dimensional risk process. Firstly, we define a one-dimensional risk model where ai> 0,i= 1,2, and∑i=1 2 ai= 1.So the two-dimensional risk model defined in Chapter 2 is changed into the one-dimensional form Then we get the Lundberg inequality and the infinite-time ruin probability using some results of the one-dimensional risk process. Secondly, we obtain the integral equation of the Gerber-Shiu discounted penalty function fC(c1)fC(c2)…fC(cn)dc1dc2…dcn and the Laplace transform ofΦ(ua) if n = 1, which is Next, we verify that the result ofΦ(r) is consistent with the one-dimensional risk model’s in special cases. Finally, we discuss the extended Gerber-Shiu discounted penalty function of the two-dimensional risk model with random premium income更多还原