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On the extension property of dilatation monotone risk measures

Author

Listed:
  • Rahsepar Massoomeh

    (Department of Mathematics, Ryerson University, Toronto, ON M5B 2K3, Canada)

  • Xanthos Foivos

    (Department of Mathematics, Ryerson University, Toronto, ON M5B 2K3, Canada)

Abstract

Let 𝒳 be a subset of L1L^{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⁢(L1,L)\sigma(L^{1},\mathcal{L}) lower semicontinuous and dilatation monotone functional ρ¯:L1→(-∞,∞]\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 L1L^{1} that retains robust representations.

Suggested Citation

  • Rahsepar Massoomeh & Xanthos Foivos, 2020. "On the extension property of dilatation monotone risk measures," Statistics & Risk Modeling, De Gruyter, vol. 37(3-4), pages 107-119, July.
    • Handle: RePEc:bpj:strimo:v:37:y:2020:i:3-4:p:107-119:n:2
    DOI: 10.1515/strm-2020-0006
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