The randomly distorted Choquet integrals with respect to a G-randomly distorted capacity and risk measures

Abstract

We study randomly distorted Choquet integrals with respect to a capacity c on a measurable space (Ω,F), where the capacity c is distorted by a G-measurable random distortion function (with G a sub-σ-algebra of F). We establish some fundamental properties, including the comonotonic additivity of these integrals under suitable assumptions on the underlying capacity space. We provide a representation result for comonotonic additive conditional risk measures which are monotone with respect to the first-order stochastic dominance relation (with respect to the capacity c) in terms of these randomly distorted Choquet integrals. We also present the case where the random distortion functions are concave. In this case, the G-randomly distorted Choquet integrals are characterised in terms of comonotonic additive conditional risk measures which are monotone with respect to the stop-loss stochastic dominance relation (with respect to the capacity c). We provide examples, extending some well-known risk measures in finance and insurance, such as the Value at Risk and the Average Value at Risk.