A note on continuity and asymptotic consistency of measures of risk and variability

Niushan Gao and Foivos Xanthos

ASTIN Bulletin, 2025, vol. 55, issue 1, 168-177

Abstract: In this short note, we show that every convex, order-bounded above functional on a Fréchet lattice is automatically continuous. This improves a result in Ruszczyński and Shapiro ((2006) Mathematics of Operations Research 31(3), 433–452.) and applies to many deviation and variability measures. We also show that an order-continuous, law-invariant functional on an Orlicz space is strongly consistent everywhere, extending a result in Krätschmer et al. ((2017) Finance and Stochastics 18(2), 271–295.).

Date: 2025
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