Compact Dual Representations and Coherent Risk Measures

Summary

This question focuses on the dual representation of coherent risk measures. It states that such a measure can be expressed as the supremum of expected losses over a set of probability measures, then asks whether an example is known whose representing set is compact and whose members are all dominated by a single reference probability measure. A closed-form expression for the risk measure is presented as an optional additional property.

The topic connects axiomatic risk measurement with properties of the set over which the dual optimization is taken. Compactness and common domination are the specific mathematical conditions under consideration, while an explicit formula would make the measure easier to calculate. The document provides no candidate measure, proof, assumptions on the underlying space, or answer about whether all requested properties can hold together. It is therefore a focused theoretical question, not an example or demonstration of a particular risk measure.

Key ideas

• A coherent risk measure can be represented through a supremum of expected losses under a family of probability measures.
• The question asks whether the representing family can be compact and dominated by one reference probability measure.
• A closed-form expression is raised as a possible additional property.
• The document gives no example or proof establishing that these conditions are jointly satisfied.

Tags

# Example of Coherent Risk measure with Compact Representation


# Example of Coherent Risk measure with Compact Representation












Every coherent risk measure $\rho$ can be represented as $$ \rho(X)\triangleq \sup_{Q \in \mathcal{Q}} \mathbb{E}\left[ -X \right], $$ for a set of probability measures $\mathcal{Q}$ defined on the same measurable space. Many of which, we know closed-form expression for (ie.: no sup, inf, or limits).

My question is, is there a known coherent risk-measure such that

- $\rho$ is coherent

- $\mathcal{Q}$ is compact in the set of probability measures, all dominated by a single probability measure $\mathbb{P}$.

- (Optional) $\rho$ has a known closed-form?

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