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
Reference25 articles.
1. C. D. Aliprantis and K. C. Border,
Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd ed.,
Springer, Berlin, 2006.
2. S. Chen, N. Gao and F. Xanthos,
The strong Fatou property of risk measures,
Depend. Model. 6 (2018), no. 1, 183–196.
3. P. Cheridito and T. Li,
Dual characterization of properties of risk measures on Orlicz hearts,
Math. Financ. Econ. 2 (2008), no. 1, 29–55.
4. P. Cheridito and T. Li,
Risk measures on Orlicz hearts,
Math. Finance 19 (2009), no. 2, 189–214.
5. A. S. Cherny and P. G. Grigoriev,
Dilatation monotone risk measures are law invariant,
Finance Stoch. 11 (2007), no. 2, 291–298.
Cited by
6 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献