Rough weak solutions for singular Lévy SDEs

Abstract

We introduce a weak solution concept (called “rough weak solutions") for singular SDEs with additive α-stable Lévy noise (including the Brownian noise case) and prove its well-posedness and equivalence to martingale solutions from Kremp and Perkowski (Bernoulli 28(3):1757–1783, 2022. https://doi.org/10.3150/21-BEJ1394) in “Young” and “rough” regularity regimes. In the rough regime this requires to construct certain rough integrals with the help of the stochastic sewing lemma, which we use to prove a generalized Itô formula for rough weak solutions. Furthermore, we show that in the Young case our solutions are equivalent to a simpler notion of weak solution, while in the rough case this simpler formulation leads to non-uniqueness in law.