We investigate the regularizing effect of certain additive continuous perturbations on SDEs with multiplicative fractional Brownian motion (fBm). Traditionally, a Lipschitz requirement on the drift and diffusion coefficients is imposed to ensure existence and uniqueness of the SDE. We show that suitable perturbations restore existence, uniqueness and regularity of the flow for the resulting equation, even when both the drift and the diffusion coefficients are distributional, thus extending the program of regularization by noise to the case of multiplicative SDEs. Our method relies on a combination of the nonlinear Young formalism developed by Catellier and Gubinelli (Stochastic Process. Appl. 126 (2016) 2323–2366), and stochastic averaging estimates recently obtained by Hairer and Li (Ann. Probab. 48 (2020) 1826–1860).
L. Galeati is funded by the DFG under Germany’s Excellence Strategy—GZ 2047/1, project-id 390685813.
F. Harang is gratefully acknowledging the financial support from the Research Council of Norway (RCN). Project STORM, project number: 274410.
The authors would like to thank the anonymous referees for their constructive comments that improved the quality of this paper.
"Regularization of multiplicative SDEs through additive noise." Ann. Appl. Probab. 32 (5) 3930 - 3963, October 2022. https://doi.org/10.1214/21-AAP1778