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arXiv · 2307.03447

Dynamic Return and Star-Shaped Risk Measures via BSDEs

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

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Roger J. A. Laeven, Emanuela Rosazza Gianin, Marco Zullino. 2023-07-07. Dynamic Return and Star-Shaped Risk Measures via BSDEs. https://arxiv.org/abs/2307.03447

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