Regularization of multiplicative SDEs through additive noise

TL;DR

This paper demonstrates that additive noise perturbations can regularize multiplicative SDEs driven by fractional Brownian motion, enabling well-posedness even with distributional coefficients, thus extending regularization by noise theory.

Contribution

It extends the regularization by noise framework to multiplicative SDEs with distributional coefficients using advanced non-linear Young and stochastic averaging techniques.

Findings

Additive noise restores existence and uniqueness of solutions.

Regularity of the flow is achieved despite distributional coefficients.

The approach combines non-linear Young formalism with stochastic averaging estimates.

Abstract

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 non-linear Young formalism developed by Catellier and Gubinelli, and stochastic averaging estimates recently obtained by Hairer and Li.