On the extension property of dilatation monotone risk measures

Massoomeh Rahsepar; Foivos Xanthos

doi:10.1515/strm-2020-0006

Article

On the extension property of dilatation monotone risk measures

Massoomeh Rahsepar and Foivos Xanthos

Published/Copyright: December 17, 2020

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From the journal Statistics & Risk Modeling Volume 37 Issue 3-4

Abstract

Let 𝒳 be a subset of L 1 that contains the space of simple random variables ℒ and ρ : X → ( - ∞ , ∞ ] a dilatation monotone functional with the Fatou property. In this note, we show that 𝜌 extends uniquely to a σ ⁢ ( L 1 , L ) lower semicontinuous and dilatation monotone functional ρ ¯ : L 1 → ( - ∞ , ∞ ] . Moreover, ρ ¯ preserves monotonicity, (quasi)convexity and cash-additivity of 𝜌. We also study conditions under which ρ ¯ preserves finiteness on a larger domain. Our findings complement extension and continuity results for (quasi)convex law-invariant functionals. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to L 1 that retains robust representations.

Keywords: Extension of risk measures; dilatation monotonicity; law invariance; Fatou property; Orlicz spaces; transformed norm risk measures; higher order dual risk measures; dual representations; Kusuoka representations

MSC 2010: 91B30; 91G80; 46E30

Funding source: Natural Sciences and Engineering Research Council of Canada

Award Identifier / Grant number: RGPIN-2019-05518

Funding statement: This research was supported by an NSERC Discovery Grant (RGPIN-2019-05518). This first author is also supported by an Ontario Graduate Scholarship.

Acknowledgements

The authors would like to thank Niushan Gao for valuable comments that improved the exposition of the manuscript. We also thank an anonymous reviewer for the insightful observations that led to include the discussion on the finiteness of ρ ¯ .

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Received: 2020-02-26

Revised: 2020-11-14

Accepted: 2020-11-30

Published Online: 2020-12-17

Published in Print: 2021-11-01

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https://doi.org/10.1515/strm-2020-0006

Keywords for this article

Extension of risk measures; dilatation monotonicity; law invariance; Fatou property; Orlicz spaces; transformed norm risk measures; higher order dual risk measures; dual representations; Kusuoka representations