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Dynamic Return and Star-Shaped Risk Measures via BSDEs

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

Abstract

This paper establishes characterization results for dynamic return and star-shaped risk measures induced via backward stochastic differential equations (BSDEs). We first characterize a general family of static star-shaped functionals in a locally convex Fr\'echet lattice. Next, employing the Pasch-Hausdorff envelope, we build a suitable family of convex drivers of BSDEs inducing a corresponding family of dynamic convex risk measures of which the dynamic return and star-shaped risk measures emerge as the essential minimum. Furthermore, we prove that if the set of star-shaped supersolutions of a BSDE is not empty, then there exists, for each terminal condition, at least one convex BSDE with a non-empty set of supersolutions, yielding the minimal star-shaped supersolution. We illustrate our theoretical results in a few examples and demonstrate their usefulness in two applications, to capital allocation and portfolio choice.

Suggested Citation

  • Roger J. A. Laeven & Emanuela Rosazza Gianin & Marco Zullino, 2023. "Dynamic Return and Star-Shaped Risk Measures via BSDEs," Papers 2307.03447, arXiv.org, revised Jul 2023.
    • Handle: RePEc:arx:papers:2307.03447
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    References listed on IDEAS

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    Cited by:

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    4. Mucahit Aygun & Roger J. A. Laeven & Mitja Stadje, 2025. "Higher-Order Ambiguity Attitudes," Papers 2501.13143, arXiv.org.
    5. Bingchu Nie & Dejian Tian & Long Jiang, 2024. "Set-valued Star-Shaped Risk Measures," Papers 2402.18014, arXiv.org, revised Feb 2025.

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