Affiliation:
1. Department of Mathematics , Ryerson University , Toronto , ON M5B 2K3 , Canada
Abstract
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.
Funder
Natural Sciences and Engineering Research Council of Canada
Subject
Statistics, Probability and Uncertainty,Modeling and Simulation,Statistics and Probability
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Risk measures on Orlicz hearts,
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Dilatation monotone risk measures are law invariant,
Finance Stoch. 11 (2007), no. 2, 291–298.
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