Random distortion risk measures

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-σ-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-σ-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-σ-algebra by focusing on the sample statistics, empirical distributions, and tail behavior of the random distortion risk measures.