On conditional distortion risk measures under uncertainty

TL;DR

This paper introduces a new class of conditional distortion risk measures that incorporate model uncertainty via an auxiliary random variable, providing a flexible framework with theoretical properties and practical comparisons.

Contribution

It proposes a novel risk measure framework under model uncertainty using an auxiliary random variable, with axiomatic characterization, coherence analysis, and dual representation.

Findings

The new risk measure framework is flexible and adaptable to various contexts.

The proposed measures are coherent and have dual representations.

Comparisons show advantages over existing risk measures like WVaR and RVaR.

Abstract

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.