【作者】 赵峰；

【导师】 高建伟；

【作者基本信息】 华北电力大学(北京) ， 管理科学与工程， 2019， 博士

【摘要】 随机占优方法为风险资产的选择提供了一个简单而有效的工具,它不需要对风险资产收益的分布、投资者需要规避的风险因子以投资者的效用函数做任何假设,而只需要比较风险资产的累积分布函数,就可以对风险资产进行排序。利用随机占优方法,将拟投资的资产划分为有效集和无效集,投资者只需在有效集中进行选择的思想,已经成为不确定性条件下金融资产投资的主要决策思想和方法之一。对随机变量的变换进行排序,是随机占优理论及应用研究的一个重要分枝。Levy于1992年提出了一般变换的随机占优判定方法,标志着关于变换型随机占优的研究基本完善。然而经过推导证明发现,Levy所给出的一般变换的二阶随机占优判定方法是错误的,甚至会得出与实际情形完全相反的结论。换言之,现有文献关于变换型随机占优关系的研究,存在一定的理论缺陷。针对变换型随机占优关系研究中的缺陷和不足,本论文主要做了以下工作:(1)通过理论分析及反例,论证了 Levy关于一般变换的二阶随机占优关系判定方法存在错误;指出对于没有任何限制条件的最一般变换,其随机占优关系无法通过变换函数和原始变量的密度函数来刻画;通过赋予变换单调性,给出了同一变量的不同变换之间随机占优关系的充分条件。(2)论证了变换具有单调性是研究变换型随机占优准则的必要前提;进而分单调递增和单调递减两种情形,分别给出连续型随机变量单调变换的随机占优准则。研究变换型随机占优的目标,是得出没有任何限制条件的一般变换的随机占优判定方法;但此前文献只给出了单调递增且连续可微变换的随机占优判定方法。通过构造若干算例并对其进行深入分析,首先得出变换函数具有单调性是研究变换型随机占优判定问题的必要前提;进而,从最基本的期望效用理论出发,考虑单调递增变换和单调递减变换两种情形,分别给出了判定连续型随机变量单调变换随机占优关系的充要条件。(3)提出离散型随机变量的变换型随机占优问题,并建立了离散型随机变量的单调变换的随机占优准则。现有文献关于变换型随机占优的研究均集中于连续型随机变量,但现实生活存在着大量的离散型变量;而且在使用计算机进行计算分析时,连续型变量也都需要进行离散化处理。更重要的是,连续型变量的变换型随机占优准则无法直接推广到离散情形。基于上述考虑,本论文提出了离散型随机变量的变换的随机占优问题,并利用变换函数,结合原始变量的概率分布,给出了离散型随机变量的单调变换的随机占优准则。(4)将变换型随机占优准则推广到几乎随机占优情形,得出变换型几乎随机占优准则。几乎随机占优是随机占优的进一步推广,具有非常广泛的应用前景,已渐渐成为随机占优理论及应用研究的一个新热点。针对随机变量的变换的随机占优问题,将其划分为连续型和离散型两种情形,并分别给出其单调变换的几乎随机占优判定准则。(5)提出一种基于变换型随机占优准则的新的随机占优判定方法。现有的随机占优判定方法,只有累积分布函数方法、分位数方法及变换型随机占优判定方法等基本方法。本论文通过证明任意的随机变量,均可以表示为连续型随机变量的单调变换,从而提出适用于普通随机变量的、基于变换型随机占优准则的一种新的随机占优判定方法,并指出分位数方法是这种新方法的一个特例。利用该方法,分析了正态分布和对数正态分布的随机占优关系、随机变量和变换之间的随机占优关系,并给出离散型随机变量的一种基于标准变换函数的新的随机占优判定方法。(6)利用变换函数建立了保险决策的数学模型,分析了变换型随机占优准则在保险、期权策略以及投资决策中的应用。此外,从风险角度出发,将变换型随机占优的原理应用于随机序和停止损失序,给出了风险变换的随机序和停止损失序的基于变换函数和原始变量概率分布的新的判定方法,并将其应用于保险策略选择。更多还原

【Abstract】 Stochastic dominance(SD)approach provides a simple and useful tool for the selection of risk assets.It can rank the risk assets without any assumptions about the utility functions and the risk factors to be avoided of the investors,and the distributions of the returns of risk assets.It is widely accepted that the risk assets for selection should first be divided into efficient and inefficient sets by the SD approach and the investor only needs to make choices in efficient sets.How to rank the transformed random variables is an important branch of the SD theory and its applications.Levy(1992)presents the SD criteria for the most general transformations,which marks that the SD theory of transformed random variables is almost perfect.However,we find that the second degree SD criteria for the most general transformation in Levy(1992)is wrong and it may lead to completely opposite results to the truth,which in turn implies that there exist significant theoretical defects in the existing literature about the SD relations between transformed random variables.In order to perfect the SD research of transformed random variables,we mainly have accomplished the following works.(1)Through theory analysis and counterexamples,we demonstrate that the second degree SD criteria for transformations in Levy(1992)is wrong,and the conditions for the first degree SD can be further relaxed.Furthermore,we point out that the monotonicity of transformation functions is indispensable for the study of SD rules for transformations,and thus present to dominance conditions for one transformation dominating another.(2)We illustrate that the monotonicity of transformation functions is necessary for the analysis of SD problem of transformed random variables.And the SD criteria for monotone transformation of continuous random variables are given.The ideal result of the study of the SD relations between transformations is to get the SD criteria for the most general transformations just as the attempt in Levy(1992),and the best result in this issue of the existing literature is the SD criteria for increasing and piecewise differentiable transformations.We first provide examples to illustrate that if the compared transformations are non-monotonic,neither the sufficient condition nor the necessary condition could be expressed by the transformation functions and the density function of the original random variable.Then,based on the expected utility theory,we divide the transformations into increasing and decreasing ones,and derive the SD criteria in both cases.Compared with the existing literature,this new SD criteria can be applied to the decreasing transformations and the non-differentiable transformations,and thus is extended to the most extensive case(3)The SD problem of discrete random variables is first proposed and the SD criteria for transformations of discrete random variables are given.To the best of our knowledge,there is no research focusing on ranking transformations under the discrete framework.It should be pointed out that the outcomes of transformations for continuous random variables cannot be extended directly to the discrete system.Notice that the discrete random variables are ubiquitous in real life and even the continuous random variables should be discretely handled in many cases,we first study the SD relations between transformation of discrete random variables,and deduce the SD criteria expressed by transformation functions and the probability function of the original random variable.(4)We extend the SD criteria for transformations to the almost SD case.To be more specific,the almost SD criteria for continuous transformations and discrete random variables are given respectively.Almost stochastic dominance is a very practical extension of the classical SD approach,and it is a hot topic in the SD theory and its applications.We present the almost SD criteria for transformations of both continuous and discrete random variables in terms of transformation functions and the probability function of the original random variable.(5)A new SD decision method for transformed random variables is given based on SD criteria for transformations.By proving that all the random variables can be induced by applying some monotonous transformations to a certain continuous random variable,we propose a new SD decision method based on SD criteria for transformations,and employ this method to rank some specific distributions.(6)By establishing relevant mathematics model,we compare the transformations resulting from the insurance and option strategies with the SD criteria for transformations.In addition,we discuss the comparing of transformations of risks,and a new judgment method of the stochastic order and the stop-loss order is proposed in terms of the transformation functions and the probability function of the original risk.With this new method,we analyze the transformations resulting from insurance strategy.更多还原