On the extension property of dilatation monotone risk measures

Abstract Let 𝒳 be a subset of L^{1} that contains the space of simple random variables ℒ and \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 \sigma(L^{1},\mathcal{L}) lower semicontinuous and dilatation monotone functional \overline{\rho}\colon L^{1}\to(-\infty,\infty] . Moreover, \overline{\rho} preserves monotonicity, (quasi)convexity and cash-additivity of 𝜌. We also study conditions under which \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} that retains robust representations.