Risk measures under model uncertainty: a Bayesian viewpoint

TL;DR

This paper explores Bayesian-inspired robust risk measures under model uncertainty, establishing conditions for their classical representation via mixture probability measures, which reflect prior beliefs and regulator perspectives.

Contribution

It introduces two types of risk measures with order and domination properties and connects them to Bayesian representations under model uncertainty.

Findings

Conditions for minimax inequality to be an equality

Representation of robust risk measures via mixture probability measures

Interpretation of Bayesian approach in risk measurement

Abstract

We introduce two kinds of risk measures with respect to some reference probability measure, which both allow for a certain order structure and domination property. Analyzing their relation to each other leads to the question when a certain minimax inequality is actually an equality. We then provide conditions under which the corresponding robust risk measures, being defined as the supremum over all risk measures induced by a set of probability measures, can be represented classically in terms of one single probability measure. We focus in particular on the mixture probability measure obtained via mixing over a set of probability measures using some prior, which represents for instance the regulator's beliefs. The classical representation in terms of the mixture probability measure can then be interpreted as a Bayesian approach to robust risk measures.