Star-shaped and dynamic return risk measures via BSDEs

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.