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<p>This paper establishes characterisation results for dynamic return and star-shaped risk measures induced via backward stochastic differential equations (BSDEs). We first characterise a general family of static star-shaped functionals on a locally convex Fréchet 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 nonempty set of supersolutions, yielding the minimal star-shaped supersolution. We illustrate our theoretical results in a few examples.</p>

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Star-shaped and dynamic return risk measures via BSDEs

  • 期刊论文

  • 发表日期：

    2026年6月17日

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

摘要

This paper establishes characterisation results for dynamic return and star-shaped risk measures induced via backward stochastic differential equations (BSDEs). We first characterise a general family of static star-shaped functionals on a locally convex Fréchet 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 nonempty set of supersolutions, yielding the minimal star-shaped supersolution. We illustrate our theoretical results in a few examples.