On the extension property of dilatation monotone risk measures

Rahsepar Massoomeh () and Xanthos Foivos ()
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Rahsepar Massoomeh: Department of Mathematics, Ryerson University, Toronto, ON M5B 2K3, Canada
Xanthos Foivos: Department of Mathematics, Ryerson University, Toronto, ON M5B 2K3, Canada

Statistics & Risk Modeling, 2020, vol. 37, issue 3-4, 107-119

Abstract: Let 𝒳 be a subset of L 1 L^{1} that contains the space of simple random variables ℒ and ρ : X → ( - ∞ , ∞ ] \rho\colon\mathcal{X}\to(-\infty,\infty] a dilatation monotone functional with the Fatou property. In this note, we show that 𝜌 extends uniquely to a σ ⁢ ( L 1 , L ) \sigma(L^{1},\mathcal{L}) lower semicontinuous and dilatation monotone functional ρ ¯ : L 1 → ( - ∞ , ∞ ] \overline{\rho}\colon L^{1}\to(-\infty,\infty] . Moreover, ρ ¯ \overline{\rho} preserves monotonicity, (quasi)convexity and cash-additivity of 𝜌. We also study conditions under which ρ ¯ \overline{\rho} 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 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 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1515/strm-2020-0006

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