On the Coherent Risk Measure Representations in the Discrete Probability Spaces

TL;DR

This paper provides a complete characterization and distinction between comonotone and not comonotone coherent risk measures in finite discrete probability spaces, using simplified AVaR representations.

Contribution

It is the first work to characterize and differentiate these risk measures via AVaR in discrete uniform probability spaces.

Findings

Complete characterization of coherent risk measures in discrete spaces.

Distinction between comonotone and not comonotone measures.

Simplified AVaR representation for applications.

Abstract

We give a complete characterization of both comonotone and not comonotone coherent risk measures in the discrete finite probability space, where each outcome is equally likely. To the best of our knowledge, this is the first work that characterizes \textit{and} distinguishes comonotone and not comonotone coherent risk measures via a simplified AVaR representation in this probability space, which is crucial in applications and simulations.