Law-Invariant Return and Star-Shaped Risk Measures
Abstract
This paper presents novel characterization results for classes of law-invariant star-shaped functionals. We begin by establishing characterizations for positively homogeneous and star-shaped functionals that exhibit second- or convex-order stochastic dominance consistency. Building on these characterizations, we proceed to derive Kusuoka-type representations for these functionals, shedding light on their mathematical structure and intimate connections to Value-at-Risk and Expected Shortfall. Furthermore, we offer representations of general law-invariant star-shaped functionals as robustifications of Value-at-Risk. Notably, our results are versatile, accommodating settings that may, or may not, involve monotonicity and/or cash-additivity. All of these characterizations are developed within a general locally convex topological space of random variables, ensuring the broad applicability of our results in various financial, insurance and probabilistic contexts.
Keywords
Cite
@article{arxiv.2310.19552,
title = {Law-Invariant Return and Star-Shaped Risk Measures},
author = {Roger J. A. Laeven and Emanuela Rosazza Gianin and Marco Zullino},
journal= {arXiv preprint arXiv:2310.19552},
year = {2023}
}