This paper presents one type of random risk measures, named as the random distortion risk measure. The random distortion risk measure is a generalization of the traditional deterministic distortion risk measure by randomizing the deterministic distortion function and the risk distribution respectively, where a stochastic distortion is introduced to randomize the distortion function, and a sub--algebra is introduced to illustrate the influence of the known information on the risk distribution. Some theoretical properties of the random distortion risk measure are provided, such as normalization, conditional positive homogeneity, conditional comonotonic additivity, monotonicity in stochastic dominance order, and continuity from below, and a method for specifying the stochastic distortion and the sub--algebra is provided. Based on some stochastic axioms, a representation theorem of the random distortion risk measure is proved. For considering the randomization of a given deterministic distortion risk measure, some families of random distortion risk measures are introduced with the stochastic distortions constructed from a Poisson process, a Brownian motion, and a Dirichlet process, respectively. A numerical analysis is carried out for showing the influence of the stochastic distortion and the sub--algebra by focusing on the sample statistics, empirical distributions, and tail behavior of the random distortion risk measures.

中文翻译：

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随机失真风险措施

本文提出了一种随机风险度量，称为随机失真风险度量。随机失真风险测度是传统确定性失真风险测度的推广，分别随机化确定性失真函数和风险分布，其中引入随机失真来随机化失真函数，并引入子代数来说明已知信息对风险分布的影响。提供了随机失真风险测度的一些理论性质，例如归一化、条件正同质性、条件共调可加性、随机优势顺序的单调性以及下连续性，​​并且指定随机失真和子代数的方法为假如。基于一些随机公理，证明了随机失真风险测度的表示定理。为了考虑给定的确定性失真风险度量的随机化，引入了一些随机失真风险度量族，其中随机失真分别由泊松过程、布朗运动和狄利克雷过程构造。通过重点关注随机失真风险度量的样本统计、经验分布和尾部行为，进行了数值分析，以显示随机失真和子代数的影响。