Dynamic Return and Star-Shaped Risk Measures via BSDEs
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.
Cite
@article{arxiv.2307.03447,
title = {Dynamic Return and Star-Shaped Risk Measures via BSDEs},
author = {Roger J. A. Laeven and Emanuela Rosazza Gianin and Marco Zullino},
journal= {arXiv preprint arXiv:2307.03447},
year = {2023}
}