On conditional distortion risk measures under uncertainty

Model uncertainty has been one prominent issue both in the theory of risk measures and in practice such as financial risk management and regulation. Motivated by this observation, in this paper, we take a new perspective to describe the model uncertainty, and thus propose a new class of risk measures under model uncertainty. More precisely, we use an auxiliary random variable to describe the model uncertainty. We first establish a conditional distortion risk measure under an auxiliary random variable. Then we axiomatically characterize it by proposing a set of new axioms. Moreover, its coherence and dual representation are investigated. Finally, we make comparisons with some known risk measures such as weighted value at risk (WVaR), range value at risk (RVaR) and $\sQ-$ mixture of ES. One advantage of our modeling is in its flexibility, as the auxiliary random variable can describe various contexts including model uncertainty. To illustrate the proposed framework, we also deduce new risk measures in the presence of background risk.This paper provides some theoretical results about risk measures under model uncertainty, being expected to make meaningful complement to the study of risk measures under model uncertainty.