Research On Stochastic Dominance Rules For Transformations And Its Insurance Decision-Making Method

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.