# On the Dual Representation of Coherent Risk Measures

arXiv: https://arxiv.org/abs/1502.06155 · HTML: https://mlsanity.com/paper/1502.06155/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2015.

## Built on by (most cited first, 8 of 8)

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- [Decision Making Needs Uncertainty Quantification [Lecture Notes]](https://mlsanity.com/paper/2607.14407/) (2026)
- [Risk Sensitivity in Markov Games and Multi-Agent Reinforcement Learning: A Systematic Review](https://mlsanity.com/paper/2406.06041/) (2024)
- [Robust Stochastic Shortest-Path Planning via Risk-Sensitive Incremental Sampling](https://mlsanity.com/paper/2408.08668/) (2024)
- [Instantiating Bayesian CVaR lower bounds in Interactive Decision Making Problems](https://mlsanity.com/paper/2604.12519/) (2026)

## Abstract (arXiv)

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.