A theory for combinations of risk measures

TL;DR

This paper develops a theoretical framework for combining risk measures without restrictive assumptions, providing dual representations and analyzing property preservation, with implications for understanding complex risk assessments.

Contribution

It introduces a novel dual representation for mixtures of convex risk measures and explores property preservation in combined risk assessment frameworks.

Findings

Derived a dual representation involving penalty functions similar to inf-convolution.

Established conditions for property preservation in combined risk measures.

Analyzed the representation of worst-case risk measures within the framework.

Abstract

We study combinations of risk measures under no restrictive assumption on the set of alternatives. We develop and discuss results regarding the preservation of properties and acceptance sets for the combinations of risk measures. One of the main results is the representation of resulting risk measures from the properties of both alternative functionals and combination functions. We build on developing a dual representation for an arbitrary mixture of convex risk measures. In this case, we obtain a penalty that recalls the notion of inf-convolution under theoretical measure integration. We develop results related to this specific context. We also explore features of individual interest generated by our frameworks, such as the preservation of continuity properties and the representation of worst-case risk measures.