两个关于相依风险的问题

发布时间：2018-04-12 10:17

本文选题：共单调 + 乘积矩　；参考：《曲阜师范大学》2015年硕士论文

【摘要】：近年来,独立风险的理论体系基本得到了完善,相依风险问题的研究在风险理论领域备受关注.在实际的生产生活中,我们必然会遇到风险相依的情况,因此测量风险、了解风险相依对破产概率的影响等问题具有非常重要的现实意义.本文考虑了两个关于相依风险的问题,一个是风险向量相依程度的测度,一个是索赔额和索赔来到的计数过程均相依情况下的破产概率的序问题.问题的解决用到了copula函数,共单调理论,超模序等工具.根据文章的具体内容,本文可分为以下两章:(1)一个基于共单调的新的多元相依测度.在这一章我们通过乘积矩的方法定义了一个基于共单调的多元相依测度,它是Koch和Schepper(ASTIN Bulletin,2011,41:191-213)以及Dhaene等(Journal of Computational and Applied Mathematics,2014,263:78-87)两篇文章改进的结果,具有较好的性质,例如它满足正则性、单调性、排列不变性以及对偶性.通过几个例子,我们对新测度与现有测度做了一下比较.当随机向量是二维时,新测度与已有测度相同,而当维数高于二维时,新测度又不同于已有的测度.最后,我们也给出了新测度的估计.(2)多维风险模型破产概率序的研究.本章我们研究了在索赔额与索赔来到过程相依的情况下,破产概率的序问题.该结果推广了Cai和Li(Journal of Multivariate Analysis,2007,98(4):757-773)的模型.在本章中,我们主要关注了三类常见的破产概率,并通过比较的方法证明了当索赔额与索赔来到过程相依程度增加时,一些破产概率如何增大,而另一些如何减小.另外,我们还根据共单调理论给出了各类破产概率的简单界.在文章最后,我们提出可以用共单调理论处理带布朗运动干扰的多维风险模型的可能性,为下一步的研究提供了一种思路.
[Abstract]:In recent years, the theoretical system of independent risk has been basically improved, and the study of dependent risk has attracted much attention in the field of risk theory.In the actual production and life, we will inevitably encounter the situation of risk dependence, so it is of great practical significance to measure the risk and understand the impact of risk dependence on the ruin probability.In this paper, we consider two problems about dependent risk, one is the measure of dependency degree of risk vector, the other is the order of ruin probability when the amount of claim and the counting process of claim are both dependent.Copula function, co-monotone theory, supermodule ordering and other tools are used to solve the problem.According to the content of this paper, this paper can be divided into the following two chapters: 1) A new multivariate dependency measure based on co-monotone.In this chapter, we define a common-monotone multivariate dependent measure based on the method of product moments, which is an improved result of two articles, Koch and Schepper(ASTIN Bulletin 2011 41: 191-213) and Dhaene et al., of Computational and Applied Mathematicsn 201443: 78-87). For example, it satisfies the regularity.Monotonicity, permutation invariance, and duality.Through several examples, we compare the new measure with the existing measure.When the random vector is two dimensional, the new measure is the same as the existing measure, and when the dimension is higher than 2 D, the new measure is different from the existing measure.Finally, we also give the estimate of the new measure.In this chapter, we study the order of ruin probability when the amount of claim is dependent on the process of claim arrival.The results extend the model of Cai and Li(Journal of Multivariate Analysis (2007).In this chapter, we mainly focus on three kinds of common ruin probability, and prove how some ruin probability increases and others decrease when the amount of claim increases with the dependence of claim coming process.In addition, we also give some simple bounds of ruin probability based on the co-monotone theory.At the end of the paper, we propose the possibility of using the co-monotone theory to deal with the multi-dimensional risk model with Brownian motion disturbance, which provides a way of thinking for the next research.
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O211.67

【共引文献】