Details
| Original language | English |
|---|---|
| Pages (from-to) | 129-141 |
| Number of pages | 13 |
| Journal | Insurance: Mathematics and Economics |
| Volume | 118 |
| Early online date | 21 Jun 2024 |
| Publication status | Published - Sept 2024 |
| Externally published | Yes |
Abstract
A functional defined on random variables f is law invariant with respect to a reference probability if its value only depends on the distribution of its argument f under that measure. In contrast to most of the literature on the topic, we take a concrete functional as given and ask if there can be more than one such reference probability. For wide classes of functionals – including, for instance, monetary risk measures and return risk measures – we demonstrate that this is not the case unless they are (i) constant, or (ii) more generally depend only on the essential infimum and essential supremum of the argument f. Mathematically, the results leverage Lyapunov's Convexity Theorem.
Keywords
- Dilatation monotonicity, Law invariance, Probabilistic sophistication, Return risk measures, Scenario-based functionals
ASJC Scopus subject areas
- Mathematics(all)
- Statistics and Probability
- Economics, Econometrics and Finance(all)
- Economics and Econometrics
- Decision Sciences(all)
- Statistics, Probability and Uncertainty
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In: Insurance: Mathematics and Economics, Vol. 118, 09.2024, p. 129-141.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Are reference measures of law-invariant functionals unique?
AU - Liebrich, Felix-Benedikt
N1 - Publisher Copyright: © 2024 The Author
PY - 2024/9
Y1 - 2024/9
N2 - A functional defined on random variables f is law invariant with respect to a reference probability if its value only depends on the distribution of its argument f under that measure. In contrast to most of the literature on the topic, we take a concrete functional as given and ask if there can be more than one such reference probability. For wide classes of functionals – including, for instance, monetary risk measures and return risk measures – we demonstrate that this is not the case unless they are (i) constant, or (ii) more generally depend only on the essential infimum and essential supremum of the argument f. Mathematically, the results leverage Lyapunov's Convexity Theorem.
AB - A functional defined on random variables f is law invariant with respect to a reference probability if its value only depends on the distribution of its argument f under that measure. In contrast to most of the literature on the topic, we take a concrete functional as given and ask if there can be more than one such reference probability. For wide classes of functionals – including, for instance, monetary risk measures and return risk measures – we demonstrate that this is not the case unless they are (i) constant, or (ii) more generally depend only on the essential infimum and essential supremum of the argument f. Mathematically, the results leverage Lyapunov's Convexity Theorem.
KW - Dilatation monotonicity
KW - Law invariance
KW - Probabilistic sophistication
KW - Return risk measures
KW - Scenario-based functionals
UR - http://www.scopus.com/inward/record.url?scp=85197382449&partnerID=8YFLogxK
U2 - 10.1016/j.insmatheco.2024.06.004
DO - 10.1016/j.insmatheco.2024.06.004
M3 - Article
VL - 118
SP - 129
EP - 141
JO - Insurance: Mathematics and Economics
JF - Insurance: Mathematics and Economics
SN - 0167-6687
ER -