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Law-Invariant Return and Star-Shaped Risk Measures

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

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.

Suggested Citation

  • Roger J. A. Laeven & Emanuela Rosazza Gianin & Marco Zullino, 2023. "Law-Invariant Return and Star-Shaped Risk Measures," Papers 2310.19552, arXiv.org.
    • Handle: RePEc:arx:papers:2310.19552
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    References listed on IDEAS

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    2. Claudia Ravanelli & Gregor Svindland, 2014. "Comonotone Pareto optimal allocations for law invariant robust utilities on L 1," Finance and Stochastics, Springer, vol. 18(1), pages 249-269, January.
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    Cited by:

    1. Yi Shen & Zachary Van Oosten & Ruodu Wang, 2024. "Partially Law-Invariant Risk Measures," Papers 2401.17265, arXiv.org.
    2. Bingchu Nie & Dejian Tian & Long Jiang, 2024. "Set-valued Star-Shaped Risk Measures," Papers 2402.18014, arXiv.org.

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