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Risk Measure Duality Without Structure
[Submitted on 8 Sep 2024 (v1), last revised 5 Aug 2026 (this ver · 2024-09-09 · via math.PR updates on arXiv.org

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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.

Submission history

From: Vasily Melnikov [view email]
[v1] Sun, 8 Sep 2024 19:12:52 UTC (21 KB)
[v2] Wed, 29 Jan 2025 21:49:15 UTC (25 KB)
[v3] Wed, 5 Aug 2026 15:52:53 UTC (16 KB)