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arXiv · 2310.19552

Law-Invariant Return and Star-Shaped Risk Measures

Roger J. A. Laeven·Emanuela Rosazza Gianin·Marco Zullino
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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.

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Roger J. A. Laeven, Emanuela Rosazza Gianin, Marco Zullino. 2023-10-30. Law-Invariant Return and Star-Shaped Risk Measures. https://arxiv.org/abs/2310.19552

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