2024/09/08 by В. А. Мельников, Vasily Melnikov, Melnikov, Vasily
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #46E30 #46N30 #91G70 #FOS: Economics and business #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Finance (q-fin.MF) #Probability (math.PR) #Risk Management (q-fin.RM) #Risk and Portfolio Optimization #Stochastic processes and financial applications #math.FA #math.PR #msc:46E30 #msc:46N30 #msc:91G70 #q-fin.MF #q-fin.RM
paper · pdf · doi:10.48550/arxiv.2409.05194
to appear in Mathematical Finance
openalex publication_date 2024/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28 · arxiv created 2026/08/05 · arxiv updated 2026/08/06
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.