On the extension property of dilatation monotone risk measures

2020/02/27 by Rahsepar, Massoomeh, Xanthos, Foivos
#FOS: Economics and business #Mathematical Finance (q-fin.MF)

paper · doi:10.48550/arxiv.2002.11865

Abstract

Let X be a subset of L¹ 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(L¹,L) lower semicontinuous and dilatation monotone functional ρ: L¹ → (-∞,∞]. Moreover, ρ preserves monotonicity, (quasi)convexity, and cash-additivity of ρ. Our findings complement recent extension results for quasiconvex law-invariant functionals proved in [17,20]. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to L¹ that retains the robust representations obtained in [4,6].

Related

• Fatou Property, representations, and extensions of law-invariant risk\n measures on general Orlicz spaces
• The strong Fatou property of risk measures
• Automatic Fatou Property of Law-invariant Risk Measures
• Law-invariant functionals on general spaces of random variables
• Law-invariant risk measures: extension properties and qualitative\n robustness