Random Distortion Risk Measures

This paper presents a random risk measure, named as the random distortionrisk measure. The random distortion risk measure is a generalization of thetraditional deterministic distortion risk measure by randomizing thedeterministic distortion function and the risk distribution respectively,where a stochastic distortion is introduced to randomize the distortionfunction, and a sub-$\sigma $-algebra is introduced for illustrating theinfluence of the known information on the risk distribution. Sometheoretical properties of the random distortion risk measure are provided,such as normalization, conditional positive homogeneity, conditionalcomonotonic additivity, monotonicity in stochastic dominance order, andcontinuity from below, and method for specifying the stochastic distortionand the sub-$\sigma $-algebra is provided. Based on some stochastic axioms,the representation theorem of the random distortion risk measure is proved.For considering the randomization of a given deterministic distortion riskmeasure, some families of random distortion risk measures are introducedwith the stochastic distortions constructed from Poisson process, Brownianmotion and Dirichlet process respectively, and numerical analysis is carriedout for showing the influence of the stochastic distortion and the riskdistribution by focusing on the sample mean, variance, skewness, kurtosis,and the tail behavior of the random distortion risk measures