本文選題：共單調 + 乘積矩　； 參考：《曲阜師范大學》2015年碩士論文

【摘要】：近年來,獨立風險的理論體系基本得到了完善,相依風險問題的研究在風險理論領域備受關注.在實際的生產(chǎn)生活中,我們必然會遇到風險相依的情況,因此測量風險、了解風險相依對破產(chǎn)概率的影響等問題具有非常重要的現(xiàn)實意義.本文考慮了兩個關于相依風險的問題,一個是風險向量相依程度的測度,一個是索賠額和索賠來到的計數(shù)過程均相依情況下的破產(chǎn)概率的序問題.問題的解決用到了copula函數(shù),共單調理論,超模序等工具.根據(jù)文章的具體內(nèi)容,本文可分為以下兩章:(1)一個基于共單調的新的多元相依測度.在這一章我們通過乘積矩的方法定義了一個基于共單調的多元相依測度,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)兩篇文章改進的結果,具有較好的性質,例如它滿足正則性、單調性、排列不變性以及對偶性.通過幾個例子,我們對新測度與現(xiàn)有測度做了一下比較.當隨機向量是二維時,新測度與已有測度相同,而當維數(shù)高于二維時,新測度又不同于已有的測度.最后,我們也給出了新測度的估計.(2)多維風險模型破產(chǎn)概率序的研究.本章我們研究了在索賠額與索賠來到過程相依的情況下,破產(chǎn)概率的序問題.該結果推廣了Cai和Li(Journal of Multivariate Analysis,2007,98(4):757-773)的模型.在本章中,我們主要關注了三類常見的破產(chǎn)概率,并通過比較的方法證明了當索賠額與索賠來到過程相依程度增加時,一些破產(chǎn)概率如何增大,而另一些如何減小.另外,我們還根據(jù)共單調理論給出了各類破產(chǎn)概率的簡單界.在文章最后,我們提出可以用共單調理論處理帶布朗運動干擾的多維風險模型的可能性,為下一步的研究提供了一種思路.
[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.
【學位授予單位】：曲阜師范大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O211.67

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本文編號：1739315