Risk Measure Duality Without Structure

Vasily Melnikov

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arXiv · 2409.05194

Risk Measure Duality Without Structure

Vasily Melnikov

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Abstract

We study risk measures on vector spaces of random variables which a priori have little structure, such as spaces lacking law invariance or a lattice structure. Ensuring the existence of a tractable dual representation (one which does not contain non-sigma-additive measures) is one of the main problems in risk measure theory, and we address it under minimal conditions. The existence of a tractable dual representation is shown to be equivalent to a Fatou-like property when the domain of the risk measure satisfies a topological regularity condition. Without the topological regularity condition, the Fatou property implies the existence of a tractable dual representation whenever the risk measure is viewed with constraints. We also present counterexamples demonstrating the sharpness of the assumptions made.

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Vasily Melnikov. 2024-09-08. Risk Measure Duality Without Structure. https://arxiv.org/abs/2409.05194

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