两个关于相依风险的问题

【作者】 张英；

【导师】 尹传存；

【作者基本信息】 曲阜师范大学 ， 概率论与数理统计， 2015， 硕士

【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.更多还原

【Abstract】 In recent years, the issue of dependent risks has received remarkable attention in the risk theory ?eld after the theoretical system of independent risks has been basically perfected.There is no doubt that we have to face the issue of dependent risks. Therefore, it’s of signi?cance to deal with the methods of measuring the dependent risks, to know how the dependence changes the ruin probability and so on. In this paper, we consider two problems about dependent risks. One is about a new dependent measure, and another one is on ordered ruin probabilities of multidimensional risk models when various types of claims are dependent and claim arrival processes are dependent at the same time. The tools we used contain copula function, comonotonicity theory, supermodular order and so on. According to the contents, the thesis can be divided into two chapters:(1) A new multivariate dependence measure based on comonotoncity.In this chapter, we introduce a new multivariate dependence measure based on comonotonicity by means of product moment which motivated by the recent papers of Koch and Schepper(ASTIN Bulletin 2011, 41: 191-213) and Dhaene et al.(Journal of Computational and Applied Mathematics 2014, 263: 78-87). The new dependence measure has many good properties including the axioms of normalization, monotonicity, permutation invariance and duality. Some difference and relation between the new dependence measure and other multivariate measures are analyzed by some examples. When m = 2 our new dependence measure is similar to the classical ones while m ≥ 3 it differs from them. In addition, we give the estimation of the new measure.(2) On ordered ruin probabilities of multidimensional risk models.In this chapter, we consider the orders of the ruin probabilities of multidimensional risk models based on the dependence among various types of claims and claim arrival processes.It generalizes the model in Cai and Li(Journal of Multivariate Analysis, 2007, 98(4):757-773). We focus on three common ruin probabilities in multidimensional compound risk models and using the comparison methods to show how some ruin probabilities increase,whereas the others decrease, as the dependence of claims and arrival processes grow. Consequently, several simple bounds are obtained based on the comonotonicity theory. The proposed method may be useful for the model perturbed by a multidimensional Brownian motion, which provides us an idea to do future work.更多还原