October 2022 Regularization of multiplicative SDEs through additive noise
Lucio Galeati, Fabian A. Harang
Author Affiliations +
Ann. Appl. Probab. 32(5): 3930-3963 (October 2022). DOI: 10.1214/21-AAP1778


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).

Funding Statement

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.


Download Citation

Lucio Galeati. Fabian A. Harang. "Regularization of multiplicative SDEs through additive noise." Ann. Appl. Probab. 32 (5) 3930 - 3963, October 2022. https://doi.org/10.1214/21-AAP1778


Received: 1 August 2020; Revised: 1 May 2021; Published: October 2022
First available in Project Euclid: 18 October 2022

MathSciNet: MR4498199
zbMATH: 07609686
Digital Object Identifier: 10.1214/21-AAP1778

Primary: 34F05 , 60H10
Secondary: 37H10

Keywords: Multiplicative noise , Regularization by noise , rough path theory , Stochastic differential equations , Young integration

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 5 • October 2022
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