Abstract
Abstract This paper presents some new stochastic dominance (SD) criteria for ranking transformations on a random variable, which is the first time that this is done for transformations under the discrete framework. By using the expected utility theory, the authors first propose a sufficient condition for general transformations by first degree SD (FSD), and further develop it into the necessary and sufficient condition for monotonic transformations. For the second degree SD (SSD) case, they divide the monotonic transformations into increasing and decreasing categories, and further derive their necessary and sufficient conditions, respectively. For two different discrete random variables with the same possible states, the authors obtain the sufficient and necessary conditions for FSD and SSD, respectively. The new SD criteria have the following features: each FSD condition is represented by the transformation functions and each SSD condition is characterized by the transformation functions and the probability distributions of the random variable. This is different from the classical SD approach where FSD and SSD conditions are described by cumulative distribution functions. Finally, a numerical example is provided to show the effectiveness of the new SD criteria.
Highlights
In real world, many human activities in insurance and financial fields induce risk transformations
With regard to the expression form, the major difference is that the existing Stochastic dominance (SD) approach is presented in the framework of cumulative distribution functions (CDFs) while the new SD rules are expressed by transformation functions and the probability function of the original random variable, so it avoids the tedious computation of CDFs and their integral
This paper developed the new first degree SD (FSD) and second degree SD (SSD) rules for ranking transformations on a discrete random variable, which is the first time to consider the ranking approach for transformations on the discrete system
Summary
Many human activities in insurance and financial fields induce risk transformations. Compared with the existing SD rules, the advantages of the new SD rules we derived are as follows: (1) the new SD rules can rank transformations on a discrete random variable, while the existing SD rules do not work; (2) the new SD rules make us avoid the tedious computation of CDFs, whereas this can not be done in the existing SD rules. In this sense, the new theoretical paradigm we derived can be regarded as a useful complement to the existing SD theory.
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