Dynamic Return and Star-Shaped Risk Measures via BSDEs

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 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 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. This speech record is from Roger's keynote speech for [2025 International Workshop on Risk Sharing (IWRS) ](https://www.keaipublishing.com/en/journals/risk-sciences/event/2025-international-workshop-on-risk-sharing-iwrs/) , to be held on July 6-7, 2025, in Beijing, China. [Risk Sciences](https://www.keaipublishing.com/en/journals/risk-sciences/) aims to foster diverse, multidisciplinary insights in risk sciences, bridging theory and practice through a series of academic activities. This talk is part of the "Research Insights" track of this series, presenting cutting-edge findings and methodologies, offering valuable perspectives to researchers and practitioners alike.