重尾论文重尾分布理论及在保险精算中的应用研究.doc

重尾论文：重尾分布理论及在保险精算中的应用研究【中文摘要】由于重尾分布能够刻画一些极端事件的损失特征,将风险模型中的索赔额约束到重尾子族,研究极端事件中保险公司的破产概率,是当前风险论研究的热点。本篇论文将重尾理论应用到风险模型中,研究索赔额随机变量属于亚指数族时,有限时间内常利力更新风险模型的破产概率渐近等价式。本文具体内容如下:第一章介绍选题的背景和本文的研究工作。第二章首先引出重尾的概念,借助一些辅助知识,系统的介绍每一子族定义及性质。重点探讨子族间的包含关系和性质,以便把重尾理论应用到以下的风险模型中。第三章以经典风险理论为起点,采用新角度从模型里的基本构造ct、S (t )推广讨论,给出各类型中具有代表性的风险模型,并根据风险模型的构造原理,介绍风险模型的研究热点。第四章假定索赔额随机变量独立同分布,其分布函数属于亚指数族,利用得到的推论,研究在常利力更新风险模型中的应用。改进以前的论证,重新证明得到有限时间内常利力更新风险模型的破产概率渐近等价式。第五章假定索赔额随机变量同分布负相依,通过推广引理,得到其分布函数属于亚指数族时的一个等价式推论。研究该等价式在改进的常利力更新风险模型中有关破产理论的应用,得到有限时间内常利力更新风险模型的破产概率渐近等价式,此结果和索赔额在独立同分布时的渐近等价式相同。第六章假定索赔额随机变量上层尾部独立。首先通过亚指数族和上层尾部独立理论的性质推广其它子族中存在的结论,然后利用该结论研究在常利力更新风险模型中,索赔额随机变量上层尾部独立且服从亚指数分布时的破产概率,得到和独立同分布时相同的破产概率渐进等价式。最后,第七章对全文进行总结分析,并在此基础上提出几个可以依据本文内容进一步展开的研究方向。【英文摘要】Many rare events can be modeled as heavy-tailed random variable. Scholars round home and abroad got some perfect results of the asymptotic estimate for the finite-time ruin probability, for the case that the random variables of the claimsizes are real-valued with common heavy-tailed distribution function, which has been a hot topic of the current risk theory research. In this paper, heavy-tailed theory will be applied to risk model. The precise asymptotic estimate for the finite-time ruin probability is established in the renewal risk model under constant interest force most by the assumption that the random variables of the claimsizes are subexponential distributions. Main contents of this dissertation are as follows:In chapter 1, the background and main research work of this dissertation are introduced.In chapter 2, the clear description of heavy-tailed is given, and then the definitions and propositions of heavy-tailed subclasses will be introduced systematically, supported from a few assistance lemmas. Since the full class of heavy-tailed distribution appears to be too big, it is necessary to research their characters and relations, which have been applied to risk model in the following chapter.In chapter 3, the clear descriptions and basic assumptions of the classical risk model are firstly given. We mainly propose a new view to extend and discuss each basic structure of the classical risk model, such asct and S (t ). Then the representative models about the classical research of risk models are presented. At last, according to the principle of risk model constructed, the trend and major research focus are mentioned.In chapter 4, for the case that the random variables are independent and real-valued with common subexponential distribution function. An equivalent formula obtained by probability theory is applied to risk model getting a deduction. Then paying particular attention to the application to ruin theory after the renewal risk model under constant interest force is proposed. Different from the old proof, the same precise asymptotic estimate for the finite-time ruin probability is established in the renewal risk model under constant interest force most by applying the former obtained deduction.In chapter 5, under the assumption that the random variables of the claimsizes are negatively dependent with common distribution function. We will extend the lemma to an equivalent formula when the claimsizes are subexponential distributions. Then paying particular attention to the application to ruin theory after the renewal risk model under constant interest force is proposed. The same precise asymptotic estimate for the finite-time ruin probability is established in the renewal risk model under constant interest force most by applying the former obtained formula. The result shows that it is still right when the claimsizes are independent with common distribution function. So it illustrates that ruin probability is insensitive to the negatively dependent structure.In chapter 6, under the assumption that the individual net losses are bivariate upper tail independent, identically distributed random variables having a common distribution in the classS . First, by the properties of the classS and upper tail independent, the conclusion is promoted existed in other class, which will be used in the renewal risk model under constant interest force to study the asymptotic estimate for the finite-time ruin probability, when the claimsizes are upper tail independent with common subexponential distribution function. The result shows that it is still right when the claimsizes are independent with common distribution function.The last chapter gives a summary of the dissertation and some possible extensions of the present work.【关键词】重尾 风险模型 独立同分布 负相依 上层尾部独立 破产概率【英文关键词】Heavy-tailed the risk model Independent and identically distribution negatively dependent upper tail independent ruin probability【目录】重尾分布理论及在保险精算中的应用研究摘要4-5ABSTRACT5-61 绪论9-121.1 选题背景9-101.2 研究工作10-122 重尾理论12-192.1 重尾子族12-162.2 重要性质16-193 风险模型19-283.1 经典风险模型19-213.1.1 模型定义