风险度量和浓度的二阶逼近以及风险厌恶的刻画
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摘要
当今社会充斥着各种各样的风险.人们一方面呼吁需要零风险社会,另一方面人类活动却增加各种各样的风险.政府部门和私人企业如工业公司,保险公司和银行都必须考虑如何应对这些风险,确保他们可以承受这些风险,不至于因为承担风险而危及到他们的机构生存.然而,在现实社会中,就像我们所看到的那样,产生致命影响的是一些极端事件的发生：如全球金融危机、经济泡沫的破灭、万年一遇的洪水、地震、火灾、风暴、飓风、火山爆发,等等.当然,我们对于风险的认知也在演变：灾难性事件已不再是过去所认为的那种不可控的和无情命运的结果.对这些事件的概率及风险度量的估计是预防、避免或至少是减少极端事件发生的不可回避且至关重要的问题.因此,迫切需要找到指导、理解和管理风险的工具.全球银行组织提出一系列的建议和规范,即著名的Basel协议Basel协会提出风险内部管理模型和与所受到的风险相对应的最小保证金的征收办法,然而一些批评家已经发现这些提议的适用性并不是很好.争论更加说明了需要更好地理解极值风险及其后果,并找到阻止或至少是最小化这种风险的办法.
基于上述所提到的问题,我们首先需要的是对风险的准确的定量认识——风险度量,其次是如何应对管理风险,如采用多样化投资使得风险分散或者贮备保证金以应对极端风险,等等.
风险度量在经济学、决策论、精算学、金融、保险和再保险等中都是一个至关重要的概念.它给人们提供了对风险的一个认识(不同的风险度量提供不同方面的信息),指导经济主体行为.从1970年起,风险度量就以保费的名义被广泛地研究.风险度量往往是基于一个公理化的方法得到的,并不存在最优的风险度量,基于不同的公理就会得到不同的风险度量,进而得到不同的风险模型.到目前为止,文献中已出现很多不同的风险度量：如广为使用的在险价值(Value-at-Risk, VaR),条件尾期望(Conditional Tail Expectation, CTE),期望短缺(Expected Shortfall),零效用保费(Zero-utility Premium), Esscher风险度量,Haezendonck-Goovaerts风险度量,等等.风险模型则有经典的期望效用模型,Yaari(1987)的对偶理论模型,更有基于两者提出的秩相依期望效用模型,等.在完全信息下,不同的行为主体对于同一风险会有不同的感知,进而会有不同的行为.这是因为影响风险度量的因素还有行为主体对待风险的态度：风险厌恶,风险中性,风险喜好.因此,在研究风险度量时,我们有必要研究行为主体对待不同风险态度的刻画.
研究表明极端事件往往具有重尾的性质,即近似地可以用生存函数满足正则变差性质的随机变量来刻画.这样我们对于极端风险的研究就转化为对那些生存函数具有正则变差性的随机变量的研究.正则变差生存函数与极值分布吸引场的关系是众所周知的且非常优美的一个结果,这也是正则变差生存函数能刻画极端事件的理论依据.
正则变差函数的概念是Jovan Karamata于1930年提出的,指的是其尾部性状类似于幂函数形式.正则变差函数有大量的优美的性质,它是人们研究极端事件的一个强有力工具.在正则变差的假设(这是合理的假设)下,可以建立极端事件风险度量的渐近等价刻画.但是,两个正则指数相同的生存函数,其尾部性状可能相差很大,此时上述的风险度量等价刻画可能会导致比较大的偏差,有时这种差异甚至可以是趋于无穷的.这促使人们去引进和研究二阶正则变差函数,并在此二阶正则变差性的假设下,建立尾事件风险度量或基于风险度量的其它特征指标(如风险浓度)的更精确的近似——二阶逼近.
本篇学位论文旨在研究风险度量和基于风险度量的风险浓度在二阶正则条件下的二阶逼近,以及在秩相依期望效用模型下若干风险厌恶概念的刻画。具体工作如下.
第二章：正则变差(RV)函数有非常多的优美性质,其中广泛应用于极值理论中的结果有Karamata定理、Breiman定理、Drees型不等式等.文献中在一般正则变差函数的基础上,提出了延展正则变差函数(ERV),二阶正则变差(2RV)函数和二阶延展正则变差函数(2ERV)的概念.后面两个概念统称为二阶正则条件.这些扩展的正则变差函数在极值理论的研究中有着非常重要的作用,这在后面会详细说明.在正则指数非零时,ERV和RV的关系是众所周知的,故几乎所有正则变差的性质都可以平移到ERV函数上de Haan&Stadtmuller (1996)建立了2ERV和ERV两者之间的联系Hua&Joe (2011)给出了2RV在一阶指数大于0,二阶指数小于0时可以转化为ERV. Neves (2009)讨论了在特定的几种情形下2ERV和2RV之间的联系.我们将在节2.3遍历所有情形,建立2ERV和2RV之间的关系.另一方面,Drees型不等式在建立尾事件概率的界和某些估计的收敛速度中有非常重要的作用.由于ERV,2RV和2ERV这三种推广的正则变差函数的定义依赖于辅助函数,Drees型不等式亦如此,且该不等式仅对特定的辅助函数成立.这样的Drees型不定式的适用范围比较窄.在节2.4中,我们给出对任意辅助函数都成立的Drees型不等式.渐近光滑是另一个重要的概念,渐近光滑性等价于规范的正则变差性,蕴含正则变差性.渐近光滑性给我们研究聚合风险的尾概率和一些风险度量的二阶展开带来方便.我们在节2.5中进一步研究渐近光滑函数的一些性质.
第三章：无论是保险公司、银行、期货公司,它们都面临很多独立的且相似的风险,为进行有效的风险管理,我们有必要研究聚合风险的性质Bar be&McCormick (2005)给出在边际风险生存函数渐近光滑的条件下,聚合风险的生存函数的二阶逼近.在险价值VaR和条件尾期望CTE是两个最常用的风险度量,有关极值风险的这两种风险度量的研究对规避风险有指导作用Degen et al.(2010)利用这一结果在渐近光滑和二阶正则变差性同时满足的假设下得到了聚合风险的VaR的二阶逼近,进而得到基于VaR的风险浓度的二阶逼近.节3.2利用二阶正则变差随机变量的CTE和VaR的渐近关系,建立了基于CTE的风险分散化效用的二阶逼近.由于渐近光滑的定义形式的复杂性,难以验证,所以我们在节3.3去除渐近光滑性的假设,仅仅在二阶正则变差的条件下,给出聚合风险的生存函数的二阶逼近、二阶正则变差性以及基于VaR和基于CTE的风险浓度的二阶逼近.这些对于风险管理和控制起着很好的指导作用.
第四章：极值分布的极大吸引场有三类,分别是Frechet类,Weibull类,Gumbel类.这三类吸引场是我们研究极端事件的具体对象,其中Frechet类和Weibull类可以用正则变差函数来刻画,Gumbel类可以用Ⅱ类来刻画.Haezendonck-Goovaerts风险度量是基于Orlicz范数而引入的保费计算准则,最初是由J.Haezendonck和M.Goovaerts于1982年提出的.Haezendonck-Goovaerts风险度量是通过一个递增且凸的Young函数来定义的.当Young函数为幂函数时,Tang&Yang(2012)分别对属于极值分布的三大极大吸引场的分布或风险变量,给出了Haezendonck-Goovaerts风险度量的一阶逼近.考虑到极值分布的三大吸引场中分布可以用其尾分位点函数(即生存函数倒数的逆函数)的ERV性质来统一刻画,我们在节4.3给出Tang&Yang关于Haezendonck-Goovaerts风险度量一阶逼近结果的新证明.新的证明利用尾分位点函数的ERV性质把三种吸引场情形统一在一起,这也有助于人们认识Haezendonck-Goovaerts风险度量.另外,我们发现对于Gumbel类,在尾分位点函数具有ERV性质的假设下,可以得到该风险度量的二阶逼近；而在尾分位点函数具有2ERV性质的假设下,节4.4分别对Frechet和Weibull吸引场中的分布建立了Haezendonck-Goovaerts风险度量的二阶逼近.节4.5给出几个具体例子说明文中所导出的Haezendonck-Goovaerts风险度量的二阶逼近的精确度远远高于一阶逼近.
第五章：风险度量的确定是依赖于决策者对于风险的态度.风险厌恶在经济学中是一个非常重要的概念,它使得决策主体避免不确定性(即风险),免于受到不可预测的事件的危害.文献中提出了很多风险厌恶的概念,其中最主要的是弱风险厌恶(Arrow,1974; Pratt,1964)、强风险厌恶(Diamond&Stiglitz,1974; Rothschild&Stigiltz,1970)、单调风险厌恶(Quiggin,1992; Landsberger&Meilijson,1994a)、左单调风险厌恶(Jewitt,1989)和右单调风险厌恶(Chateauneuf et al.,2004)在经典的期望效用模型下,这五种风险厌恶的概念是等价的.由Allais悖论,我们知道期望效用模型并不能很好地解释所有的经济行为,这是因为在期望效用模型下,每个决策者都是被一个效用函数所刻画,这个函数既刻画决策者对财富的态度,也刻画其对不确定性的态度,即每个决策者是边际效应递减的同时一定是厌恶风险的.然而,事实上大部分的人愿意购买保险的同时也愿意购买彩票,这就说明了某些决策者厌恶风险的同时也可以是追逐巨额财富的.秩相依期望效用模型就是鉴于此提出的.在秩相依期望效用模型中,每个决策者被两个函数所刻画——效用函数u和概率感知函数f.用效用函数和概率感知函数来刻画几种风险厌恶的概念是一个非常有趣且有意义的工作.在限制我们所面临的风险都是有界的条件下,Chew et al.(1987)和Ryan(2006)证明强风险厌恶是等价于u是凹函数且f是凸函数；Chateauneuf et al(2005)引入了两种指数——贪婪指数(Q)和悲观指数(P),这两种指数分别是通过效用函数和概率感知函数来定义的,他们证明了单调风险厌恶是等价于贪婪指数Q小于或等于悲观指数P; Ryan (2006)对于左单调风险厌恶通过定义新的悲观指数给出了类似的刻画(虽然结果是对的,但是最主要定理的证明中存在明显的错误).另一方面,现实生活中无界风险变量随处可见,一个自然的问题是上述的风险厌恶的刻画对无界风险变量是否依然正确?这是一个非常有意义且富于挑战性的问题.正如Rothschild&Stigiltz (1970)中所说的那样,将某些问题从有界随机变量推广到无界随机变量不是一件简单的工作,这需要涉及一系列精细的收敛问题.在这里,通常的弱收敛的概念已经不够了,需要引进和使用新的收敛概念.在第五章中,我们首先给出有界风险变量的左单调风险厌恶的刻画,纠正Ryan(2006)中的证明错误,然后通过截尾的方法(截尾的同时一定要保证风险变量之间特定的随机序关系依然成立)将上述四种风险厌恶的刻画都推广到无界风险变量.
第六章：风险度量是基于一系列公理提出的,基于不同的公理可以得到不同风险度量.例如,von Neumann&Morgenstern(1947)给出了期望效用模型基于一组公理的刻画(或参考Fishburn,1982),若将其中的独立性公理替换为同单调性公理,则我们可以得到Yaari对偶理论,亦即扭曲风险度量.同单调是经济学和精算学中一个非常重要的概念,指的是一组风险变量之间的关系是同增同减的.对于任意n个风险变量X1,….,Xn,我们用X1c,….,Xnc表示相应的同单调版本的风险变量,则其聚合风险∑in=1Xi在停止损失序(或凸序)意义下不大于同单调版本的聚合风险∑i=1n Xic(Kaas et al,2002)在所考虑的概率空间是无原子的假设下,Chueng (2008,2010)证明了上述结果的逆命题成立,即如果一组风险变量之和在停止损失序意义下不小于任意一组有相同边际分布的风险变量之和,那么这组风险之间的关系是同单调的.他的证明非常繁琐复杂.在第六章中,我们取消概率空间无原子的假设,用非常简单明了的方法证明了该结果成立.
本学位论文的主要创新点如下：
1.基于ERV、2RV和2ERV的经典Drees型不等式,证明了对任意辅助函数都成立的更为一般的Drees型不等式,该不等式具有广泛的适用性；同时深入探讨2RV和2ERV之间的蕴涵关系.
2.证明了Degen et al(2010)在渐近光滑和二阶正则性的条件下得到的基于VaR的风险浓度的二阶逼近结果在仅有二阶正则条件下依然成立
3.利用尾分位点函数的ERV性完全刻画三大极值分布吸引场,将Tang&Yang (2012)分别针对三大极值分布吸引场建立的Haezendonck-Goovaerts风险度量的一阶逼近纳入到统一的模式中.更进一步地,在条件不变的情况下,对Gumbel吸引场给出Haezendonck-Goovaerts风险度量的二阶逼近,并在二阶正则条件下给出Frechet, Weibull分布极大吸引场对应的Haezendonck-Goovaerts风险度量的二阶逼近.
4.在秩相依的期望效用模型框架下,对无界风险变量建立了左单调风险厌恶和右单调风险厌恶的刻画,纠正了Ryan (2006)关于有界风险变量左单调风险厌恶刻画证明中的错误,并将Chew et al(1987)关于强风险厌恶,Chateauneuf et al.(2005)关于单调风险厌恶的刻画从有界风险拓展到到无界风险变量.这一研究富有挑战性,有相当的难度.我们找到了解决问题的有效方法,这就是从微观层面去研究随机序,揭示随机序内在的深层次的性质.从宏观层面转向从微观层面研究随机序也代表了未来随机序理论研究的一个主流方法.
5.用简洁有趣的方法证明了Cheung(2010)关于同单调相依风险的等价刻画,并取消了底概率空间无原子的假设.
Regular variation (RV) has become one of the key notions which appears in a natural way in applied probability, statistics, risk management, and other fields. There are a variety of concepts extending RV, among which are the ex-tended regular variation (ERV), second-order regular variation (2RV) and second-order extended regular variation (2ERV). Here,2RV and2ERV are termed as the second-order conditions. The quantification of diversification benefits due to risk aggregation has received more attention in the recent literature. Because risk managers become more and more concerned with the tail area of risks due to the excessive prudence of regulatory framework, there is an urgent need to establish second-order approximations of some risk measures and risk concentrations for extreme tails. The second-order condition provides a platform to do such a study.
Risk aversion is a crucial concept in the economics. Risk aversion is the atti-tude that induces people to avoid uncertainty, to be protected from unpredictable events, and to buy some financial products. Various notions of risk aversion have been introduced in the literature, for example, the weak risk aversion, the strong risk aversion, the monotone risk aversion, the left-monotone risk aversion, and the right-monotone risk aversion. There are some literatures on characterizing these notions of risk aversion in the framework of the rank-dependent expected utility (RDEU) model for bounded random variables. It is interesting to identify sufficient and/or necessary conditions on the RDEU model such that a decision maker exhibits some notion of risk aversion.
The purposes of this thesis are to study the second-order approximations of some risk measures and risk concentrations under the second-order condition, and to characterize the concepts of risk aversion in the RDEU model. The main contributions of this thesis are as follows.
1. Based on the classical Drees-type inequalities for ERV,2RV and2ERV func-tions, we establish new Drees-type inequalities with arbitrary auxiliary func-tions. This kind of inequalities has potential applications. The connections between2RV and2ERV are investigated carefully.
2. Degen et al.(2010) derived second-order approximations of the risk con-centration based on the Value-at-Risk (VaR) for iid loss variables with a common survival function possessing the properties of2RV and of asymp-totic smoothness. We remove the assumption of the asymptotic smoothness, and reestablish the second-order expansions of risk concentration based on VaR and conditional tail expectation (CTE).
3. We establish the second-order approximations of Haezendonck-Goovaerts risk measure under the2RV condition upon the tail quantile function U(t), and also reprove the main results in Tang and Yang (2012), concerning the first-order approximations of the Haezendonck-Goovaerts risk measure under the ERV condition upon U(t).
4. In the framework of the RDEU model, we characterize the left-monotone and the right-monotone risk aversions for unbounded random variables, and remove the gap in the proof of the main result in Ryan (2006) concerning the characterization of the left-monotone risk aversion for bounded random vari-ables. We also extend the characterizations of the strong risk aversion and the monotone risk aversion obtained respectively by Chew et al.(1987) and Chateauneuf et al.(2005) from bounded to unbounded random variables.
5. Cheung (2010) characterized the comonotonicity of random variables in terms of the convex order, provided the underlying probability space is atomless. We remove the assumption that the probability space is atom-less, and give a new and simple proof for such a characterization.
基于上述所提到的问题,我们首先需要的是对风险的准确的定量认识——风险度量,其次是如何应对管理风险,如采用多样化投资使得风险分散或者贮备保证金以应对极端风险,等等.
风险度量在经济学、决策论、精算学、金融、保险和再保险等中都是一个至关重要的概念.它给人们提供了对风险的一个认识(不同的风险度量提供不同方面的信息),指导经济主体行为.从1970年起,风险度量就以保费的名义被广泛地研究.风险度量往往是基于一个公理化的方法得到的,并不存在最优的风险度量,基于不同的公理就会得到不同的风险度量,进而得到不同的风险模型.到目前为止,文献中已出现很多不同的风险度量：如广为使用的在险价值(Value-at-Risk, VaR),条件尾期望(Conditional Tail Expectation, CTE),期望短缺(Expected Shortfall),零效用保费(Zero-utility Premium), Esscher风险度量,Haezendonck-Goovaerts风险度量,等等.风险模型则有经典的期望效用模型,Yaari(1987)的对偶理论模型,更有基于两者提出的秩相依期望效用模型,等.在完全信息下,不同的行为主体对于同一风险会有不同的感知,进而会有不同的行为.这是因为影响风险度量的因素还有行为主体对待风险的态度：风险厌恶,风险中性,风险喜好.因此,在研究风险度量时,我们有必要研究行为主体对待不同风险态度的刻画.
研究表明极端事件往往具有重尾的性质,即近似地可以用生存函数满足正则变差性质的随机变量来刻画.这样我们对于极端风险的研究就转化为对那些生存函数具有正则变差性的随机变量的研究.正则变差生存函数与极值分布吸引场的关系是众所周知的且非常优美的一个结果,这也是正则变差生存函数能刻画极端事件的理论依据.
正则变差函数的概念是Jovan Karamata于1930年提出的,指的是其尾部性状类似于幂函数形式.正则变差函数有大量的优美的性质,它是人们研究极端事件的一个强有力工具.在正则变差的假设(这是合理的假设)下,可以建立极端事件风险度量的渐近等价刻画.但是,两个正则指数相同的生存函数,其尾部性状可能相差很大,此时上述的风险度量等价刻画可能会导致比较大的偏差,有时这种差异甚至可以是趋于无穷的.这促使人们去引进和研究二阶正则变差函数,并在此二阶正则变差性的假设下,建立尾事件风险度量或基于风险度量的其它特征指标(如风险浓度)的更精确的近似——二阶逼近.
本篇学位论文旨在研究风险度量和基于风险度量的风险浓度在二阶正则条件下的二阶逼近,以及在秩相依期望效用模型下若干风险厌恶概念的刻画。具体工作如下.
第二章：正则变差(RV)函数有非常多的优美性质,其中广泛应用于极值理论中的结果有Karamata定理、Breiman定理、Drees型不等式等.文献中在一般正则变差函数的基础上,提出了延展正则变差函数(ERV),二阶正则变差(2RV)函数和二阶延展正则变差函数(2ERV)的概念.后面两个概念统称为二阶正则条件.这些扩展的正则变差函数在极值理论的研究中有着非常重要的作用,这在后面会详细说明.在正则指数非零时,ERV和RV的关系是众所周知的,故几乎所有正则变差的性质都可以平移到ERV函数上de Haan&Stadtmuller (1996)建立了2ERV和ERV两者之间的联系Hua&Joe (2011)给出了2RV在一阶指数大于0,二阶指数小于0时可以转化为ERV. Neves (2009)讨论了在特定的几种情形下2ERV和2RV之间的联系.我们将在节2.3遍历所有情形,建立2ERV和2RV之间的关系.另一方面,Drees型不等式在建立尾事件概率的界和某些估计的收敛速度中有非常重要的作用.由于ERV,2RV和2ERV这三种推广的正则变差函数的定义依赖于辅助函数,Drees型不等式亦如此,且该不等式仅对特定的辅助函数成立.这样的Drees型不定式的适用范围比较窄.在节2.4中,我们给出对任意辅助函数都成立的Drees型不等式.渐近光滑是另一个重要的概念,渐近光滑性等价于规范的正则变差性,蕴含正则变差性.渐近光滑性给我们研究聚合风险的尾概率和一些风险度量的二阶展开带来方便.我们在节2.5中进一步研究渐近光滑函数的一些性质.
第三章：无论是保险公司、银行、期货公司,它们都面临很多独立的且相似的风险,为进行有效的风险管理,我们有必要研究聚合风险的性质Bar be&McCormick (2005)给出在边际风险生存函数渐近光滑的条件下,聚合风险的生存函数的二阶逼近.在险价值VaR和条件尾期望CTE是两个最常用的风险度量,有关极值风险的这两种风险度量的研究对规避风险有指导作用Degen et al.(2010)利用这一结果在渐近光滑和二阶正则变差性同时满足的假设下得到了聚合风险的VaR的二阶逼近,进而得到基于VaR的风险浓度的二阶逼近.节3.2利用二阶正则变差随机变量的CTE和VaR的渐近关系,建立了基于CTE的风险分散化效用的二阶逼近.由于渐近光滑的定义形式的复杂性,难以验证,所以我们在节3.3去除渐近光滑性的假设,仅仅在二阶正则变差的条件下,给出聚合风险的生存函数的二阶逼近、二阶正则变差性以及基于VaR和基于CTE的风险浓度的二阶逼近.这些对于风险管理和控制起着很好的指导作用.
第四章：极值分布的极大吸引场有三类,分别是Frechet类,Weibull类,Gumbel类.这三类吸引场是我们研究极端事件的具体对象,其中Frechet类和Weibull类可以用正则变差函数来刻画,Gumbel类可以用Ⅱ类来刻画.Haezendonck-Goovaerts风险度量是基于Orlicz范数而引入的保费计算准则,最初是由J.Haezendonck和M.Goovaerts于1982年提出的.Haezendonck-Goovaerts风险度量是通过一个递增且凸的Young函数来定义的.当Young函数为幂函数时,Tang&Yang(2012)分别对属于极值分布的三大极大吸引场的分布或风险变量,给出了Haezendonck-Goovaerts风险度量的一阶逼近.考虑到极值分布的三大吸引场中分布可以用其尾分位点函数(即生存函数倒数的逆函数)的ERV性质来统一刻画,我们在节4.3给出Tang&Yang关于Haezendonck-Goovaerts风险度量一阶逼近结果的新证明.新的证明利用尾分位点函数的ERV性质把三种吸引场情形统一在一起,这也有助于人们认识Haezendonck-Goovaerts风险度量.另外,我们发现对于Gumbel类,在尾分位点函数具有ERV性质的假设下,可以得到该风险度量的二阶逼近；而在尾分位点函数具有2ERV性质的假设下,节4.4分别对Frechet和Weibull吸引场中的分布建立了Haezendonck-Goovaerts风险度量的二阶逼近.节4.5给出几个具体例子说明文中所导出的Haezendonck-Goovaerts风险度量的二阶逼近的精确度远远高于一阶逼近.
第五章：风险度量的确定是依赖于决策者对于风险的态度.风险厌恶在经济学中是一个非常重要的概念,它使得决策主体避免不确定性(即风险),免于受到不可预测的事件的危害.文献中提出了很多风险厌恶的概念,其中最主要的是弱风险厌恶(Arrow,1974; Pratt,1964)、强风险厌恶(Diamond&Stiglitz,1974; Rothschild&Stigiltz,1970)、单调风险厌恶(Quiggin,1992; Landsberger&Meilijson,1994a)、左单调风险厌恶(Jewitt,1989)和右单调风险厌恶(Chateauneuf et al.,2004)在经典的期望效用模型下,这五种风险厌恶的概念是等价的.由Allais悖论,我们知道期望效用模型并不能很好地解释所有的经济行为,这是因为在期望效用模型下,每个决策者都是被一个效用函数所刻画,这个函数既刻画决策者对财富的态度,也刻画其对不确定性的态度,即每个决策者是边际效应递减的同时一定是厌恶风险的.然而,事实上大部分的人愿意购买保险的同时也愿意购买彩票,这就说明了某些决策者厌恶风险的同时也可以是追逐巨额财富的.秩相依期望效用模型就是鉴于此提出的.在秩相依期望效用模型中,每个决策者被两个函数所刻画——效用函数u和概率感知函数f.用效用函数和概率感知函数来刻画几种风险厌恶的概念是一个非常有趣且有意义的工作.在限制我们所面临的风险都是有界的条件下,Chew et al.(1987)和Ryan(2006)证明强风险厌恶是等价于u是凹函数且f是凸函数；Chateauneuf et al(2005)引入了两种指数——贪婪指数(Q)和悲观指数(P),这两种指数分别是通过效用函数和概率感知函数来定义的,他们证明了单调风险厌恶是等价于贪婪指数Q小于或等于悲观指数P; Ryan (2006)对于左单调风险厌恶通过定义新的悲观指数给出了类似的刻画(虽然结果是对的,但是最主要定理的证明中存在明显的错误).另一方面,现实生活中无界风险变量随处可见,一个自然的问题是上述的风险厌恶的刻画对无界风险变量是否依然正确?这是一个非常有意义且富于挑战性的问题.正如Rothschild&Stigiltz (1970)中所说的那样,将某些问题从有界随机变量推广到无界随机变量不是一件简单的工作,这需要涉及一系列精细的收敛问题.在这里,通常的弱收敛的概念已经不够了,需要引进和使用新的收敛概念.在第五章中,我们首先给出有界风险变量的左单调风险厌恶的刻画,纠正Ryan(2006)中的证明错误,然后通过截尾的方法(截尾的同时一定要保证风险变量之间特定的随机序关系依然成立)将上述四种风险厌恶的刻画都推广到无界风险变量.
第六章：风险度量是基于一系列公理提出的,基于不同的公理可以得到不同风险度量.例如,von Neumann&Morgenstern(1947)给出了期望效用模型基于一组公理的刻画(或参考Fishburn,1982),若将其中的独立性公理替换为同单调性公理,则我们可以得到Yaari对偶理论,亦即扭曲风险度量.同单调是经济学和精算学中一个非常重要的概念,指的是一组风险变量之间的关系是同增同减的.对于任意n个风险变量X1,….,Xn,我们用X1c,….,Xnc表示相应的同单调版本的风险变量,则其聚合风险∑in=1Xi在停止损失序(或凸序)意义下不大于同单调版本的聚合风险∑i=1n Xic(Kaas et al,2002)在所考虑的概率空间是无原子的假设下,Chueng (2008,2010)证明了上述结果的逆命题成立,即如果一组风险变量之和在停止损失序意义下不小于任意一组有相同边际分布的风险变量之和,那么这组风险之间的关系是同单调的.他的证明非常繁琐复杂.在第六章中,我们取消概率空间无原子的假设,用非常简单明了的方法证明了该结果成立.
本学位论文的主要创新点如下：
1.基于ERV、2RV和2ERV的经典Drees型不等式,证明了对任意辅助函数都成立的更为一般的Drees型不等式,该不等式具有广泛的适用性；同时深入探讨2RV和2ERV之间的蕴涵关系.
2.证明了Degen et al(2010)在渐近光滑和二阶正则性的条件下得到的基于VaR的风险浓度的二阶逼近结果在仅有二阶正则条件下依然成立
3.利用尾分位点函数的ERV性完全刻画三大极值分布吸引场,将Tang&Yang (2012)分别针对三大极值分布吸引场建立的Haezendonck-Goovaerts风险度量的一阶逼近纳入到统一的模式中.更进一步地,在条件不变的情况下,对Gumbel吸引场给出Haezendonck-Goovaerts风险度量的二阶逼近,并在二阶正则条件下给出Frechet, Weibull分布极大吸引场对应的Haezendonck-Goovaerts风险度量的二阶逼近.
4.在秩相依的期望效用模型框架下,对无界风险变量建立了左单调风险厌恶和右单调风险厌恶的刻画,纠正了Ryan (2006)关于有界风险变量左单调风险厌恶刻画证明中的错误,并将Chew et al(1987)关于强风险厌恶,Chateauneuf et al.(2005)关于单调风险厌恶的刻画从有界风险拓展到到无界风险变量.这一研究富有挑战性,有相当的难度.我们找到了解决问题的有效方法,这就是从微观层面去研究随机序,揭示随机序内在的深层次的性质.从宏观层面转向从微观层面研究随机序也代表了未来随机序理论研究的一个主流方法.
5.用简洁有趣的方法证明了Cheung(2010)关于同单调相依风险的等价刻画,并取消了底概率空间无原子的假设.
Regular variation (RV) has become one of the key notions which appears in a natural way in applied probability, statistics, risk management, and other fields. There are a variety of concepts extending RV, among which are the ex-tended regular variation (ERV), second-order regular variation (2RV) and second-order extended regular variation (2ERV). Here,2RV and2ERV are termed as the second-order conditions. The quantification of diversification benefits due to risk aggregation has received more attention in the recent literature. Because risk managers become more and more concerned with the tail area of risks due to the excessive prudence of regulatory framework, there is an urgent need to establish second-order approximations of some risk measures and risk concentrations for extreme tails. The second-order condition provides a platform to do such a study.
Risk aversion is a crucial concept in the economics. Risk aversion is the atti-tude that induces people to avoid uncertainty, to be protected from unpredictable events, and to buy some financial products. Various notions of risk aversion have been introduced in the literature, for example, the weak risk aversion, the strong risk aversion, the monotone risk aversion, the left-monotone risk aversion, and the right-monotone risk aversion. There are some literatures on characterizing these notions of risk aversion in the framework of the rank-dependent expected utility (RDEU) model for bounded random variables. It is interesting to identify sufficient and/or necessary conditions on the RDEU model such that a decision maker exhibits some notion of risk aversion.
The purposes of this thesis are to study the second-order approximations of some risk measures and risk concentrations under the second-order condition, and to characterize the concepts of risk aversion in the RDEU model. The main contributions of this thesis are as follows.
1. Based on the classical Drees-type inequalities for ERV,2RV and2ERV func-tions, we establish new Drees-type inequalities with arbitrary auxiliary func-tions. This kind of inequalities has potential applications. The connections between2RV and2ERV are investigated carefully.
2. Degen et al.(2010) derived second-order approximations of the risk con-centration based on the Value-at-Risk (VaR) for iid loss variables with a common survival function possessing the properties of2RV and of asymp-totic smoothness. We remove the assumption of the asymptotic smoothness, and reestablish the second-order expansions of risk concentration based on VaR and conditional tail expectation (CTE).
3. We establish the second-order approximations of Haezendonck-Goovaerts risk measure under the2RV condition upon the tail quantile function U(t), and also reprove the main results in Tang and Yang (2012), concerning the first-order approximations of the Haezendonck-Goovaerts risk measure under the ERV condition upon U(t).
4. In the framework of the RDEU model, we characterize the left-monotone and the right-monotone risk aversions for unbounded random variables, and remove the gap in the proof of the main result in Ryan (2006) concerning the characterization of the left-monotone risk aversion for bounded random vari-ables. We also extend the characterizations of the strong risk aversion and the monotone risk aversion obtained respectively by Chew et al.(1987) and Chateauneuf et al.(2005) from bounded to unbounded random variables.
5. Cheung (2010) characterized the comonotonicity of random variables in terms of the convex order, provided the underlying probability space is atomless. We remove the assumption that the probability space is atom-less, and give a new and simple proof for such a characterization.
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[37]Diamond, P. and Stiglitz, J. (1974). Increases in risk and risk aversion. Journal of Economic Theory,8,337-360.
[38]Drees, H. (1998). On smooth statistical tail functionals. Scandinavian Journal of Statistics,25,187-210.
[39]Dutta, K. and Perry, J. (2006). A tale of tails:an empirical analysis of loss distribu-tion models for estimating operational risk capital. Federal Reserve Bank of Boston, Working Paper No.06-13.
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[41]Embrechts, P., Lambrigger, D.D. and Wuthrich, M.V. (2009a). Multivariare extremes and the aggregation of dependent risks:Examples and counterexamples. Extremes, 12,107-127.
[42]Embrechts, P., Neslehova, J. and Wuthrich, M.V. (2009b). Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness. Insurance: Mathematics and Economics,44,164-169.
[43]Fagiuoli, E., Pellerey, F. and Shaked, M. (1999). A characterization of the dilation order and its applications. Statistical Papers,40,393-406.
[44]Fernandez-Ponce, J.M., Kochar, S.C. and Munoz-Perez, J. (1998). Partial orderings of distributions based on right spread functions. Journal of Applied Probability,35, 221-228.
[45]Fishburn, P.C. (1982). The Foundations of Expected Utility. Reidel, Dordrecht.
[46]Fraga Alves, M.I., Gomes, M.I., de Haan, L. and Neves, C. (2007). A note on second order conditions in extreme value theory:Linking general and heavy tails conditions. Revstat,5,285-304.
[47]Galambos, J. (2001). The Asymptotic Theory of Extreme Order Statistics (Second Edition). Robert E. Krieger Publishing Go., Inc.
[48]Geluk, J.L. (1996). Tails of subordinated laws:The regularly varying case. Stochastic Processes and Their Applications,61,147-161.
[49]Geluk, J.L., de Haan, L., Resnick S. and Starica C. (1997). Second-order regular variation, convolution and the central limit theorem. Sochastic Process and Their Applications,69,139-159.
[50]Goovaerts, M., Kaas, R., Dhaene, J. and Tang, Q. (2004). Some new classes of con-sistent risk measures. Insurance:Mathematics and Economics,34,505-516.
[51]Goovaerts, M., Kaas, R. and Laeven, R.J. (2010). A note on additive risk measures in rank-dependent utility. Insurance:Mathematics and Economics,47,187-189.
[52]Goovaerts, M., Linders, D., Van Weert, K. and Tank, F. (2012). On the interplay between distortion, mean value and Haezendonck risk measures. Insurance:Mathe-matics and Economics,51,10-18.
[53]Grant, S. and Kajii, A. (1994). AUSI expected utility:An anticipated utility theory of relative disappointment aversion. CORE discussion papper 9445.
[54]Haezendonck, J. and Goovaerts, M. (1982). A new premium calculation principle based on Orlicz norms. Insurance:Mathematics and Economics,1,41-53.
[55]Hall, P. (1982). On some simple estimates of an exponent of regular variation. Journal of the Royal Statistical Society B,44,37-42.
[56]Hu, T., Chen, J. and Yao, J. (2006). Preservation of the location independent risk order under convolution. Insurance:Mathematics and Economics,38,406-412.
[57]Hua, L. and Joe, H. (2011). Second order regular variation and conditional tail ex-pectation of multiple risks. Insurance:Mathematics and Economics,49,537-546.
[58]Jewitt, I. (1989). Choosing between risky prospects:the characterization of compar-ative statics results, and location independent risk. Management Science,35,60-70.
[59]Kaas, R., Dhaene, J., Vyncke, D., Goovaerts, M. and Denuit, M. (2002). A simple geometry proof that comonotonic risks have the convex-largest sum. ASTIN Bulletin, 32,71-80.
[60]Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2008). Modern Actuarial Risk Theory:Using R (2nd edition). Springer.
[61]Kochar, S.C., Li, X. and Shaked, M. (2002). The total time on test transform and the excess wealth stochastic orders of distributions. Advances in Applied Probability, 34,826-845.
[62]Kratschmer, V. and Zahle, H. (2011). Sensitivity of risk measures with respect to the normal approximation of total claim distributions. Insurance:Mathematics and Economics,49,335-344.
[63]Landsberger, M. and Meilijson, I. (1994a). Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion. Annals of Operations Re-search,52,97-106.
[64]Landsberger, M. and Meilijson, I. (1994b). The generating process and an extension of Jewitt's location independent risk concept. Management Science,40,662-669.
[65]Lv, W., Mao, T. and Hu, T. (2011). Properties of second-order regular variation and expansions for risk concentration. Probability in the Engineering and Informational Sciences. To appear.
[66]Mainik, G. (2011). Estimatinmg asymptotic dependence functionals in multivariate regularly varying models. Submitted.
[67]Mainik, G. and Ruschendord, L. (2010). On optimal portfolio diversification with respect to extreme risks. Finance and Stochastics,14,593-623.
[68]Miiller, A. (1998). Comparing risks with unbounded distributions. Journal of Math-ematical Economics 30,229-239.
[69]Miiller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley & Sons, Ltd., West Sussex.
[70]Nam, H.S., Tang, Q. and Yang, F. (2011). Characterization of upper comonotonicity via tail convex order. Insurance:Mathematics and Economics,48,368-373.
[71]Neves, C. (2009). From extended regular variation to regular variation with applica-tion in extreme value statistics. Journal of Mathematical Analysis and Applications, 355,216-230.
[72]Peng, Z. and Nadarajah, S. (2008). Second order regular variation and its applications to rates of convergence in extreme-value distribution. Bulletin of Korean Mathematical Society,45,75-93.
[73]Pratt, J.W. (1964). Risk aversion in the small and in the large. Econometrica,32, 122-136.
[74]Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior and Organization,3,323-343.
[75]Quiggin, J. (1992). Increasing risk:another definition. In:Chikan, A. (Ed.), Progress in Decision, Utility and Risk Theory. Kluwer, Dordrecht.
[76]Resnick, S.I. (2007). Heavy-tail Phenomena. Springer Series in Operations Research and Financial Engineering. Spriger, New York.
[77]Rothschild, M. and Stiglitz, J. (1970). Increasing risk I:a definition. Journal of Eco-nomic Theory,2,225-243.
[78]Ryan, M.J. (2006). Risk aversion in RDEU. Journal of Mathematical Economics,42, 675-697.
[79]Scarsini, M. (1994). Comparing risk and risk aversion, In:Shaked, M., Shanthikumar, J.G. (Eds.), Stochastic Orders and Their Applications, Academic Press, pp.351-378.
[80]Schmidt, U. and Zank, H. (2002). Risk aversionin cumulative prospect theory, mimeo, University of Manchester.
[81]Shaked, M. and Shanthikumar, J.G. (1998). Two variability orders. Probability in the Engineering and Informational Sciences,12,1-23.
[82]Shaked, M. and Shanthikumar, J.G. (2007). Stochastic Orders. Springer, New York.
[83]Tang, Q. and Yang, F. (2012). On the Haezendonck-Goovaerts risk measure for ex-treme risks. Insurance:Mathematics and Economics,50,217-227.
[84]Tong, B., Wu, C.F. and Xu, W.D. (2012). Risk concentration of aggregate dependent risks:The second-order properties. Insurance:Mathematics and Economics,50,139-149.
[85]Von Neumann, J. and Morgenstern, O. (1947). Theory of Games and Economic Be-havior (2nd edn), Princeton University Press, Princeton, NJ.
[86]Wang, S. (1996). Premium calculation by transforming the layer premium density. ASTIN Bulletin,26,71-92.
[87]Yaari, M.E. (1987). The dual theory of choice under risk. Econometrica,55,95-115.
[88]Zhou, C. (2010). Dependence structure of risk factors and diversification effects. In-surance:Mathematics and Economics,46,531-540.