  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.
