On the Dual Representation of Coherent Risk Measures

TL;DR

This paper explores the dual representation of coherent risk measures, analyzing how risk envelopes relate to risk measures, their structure, and connections to robust optimization, providing deeper theoretical insights.

Contribution

It offers a detailed study of risk envelope operations, structures of popular risk measures, and links between stochastic and robust optimization frameworks.

Findings

Set operations affect risk measure properties

Structural analysis of risk envelopes for common measures

Connections established between risk measures and uncertainty sets

Abstract

A classical result in risk measure theory states that every coherent risk measure has a dual representation as the supremum of certain expected value over a risk envelope. We study this topic in more detail. The related issues include: 1. Set operations of risk envelopes and how they change the risk measures, 2. The structure of risk envelopes of popular risk measures, 3. Aversity of risk measures and its impact to risk envelopes, and 4. A connection between risk measures in stochastic optimization and uncertainty sets in robust optimization.