Weak solutions for singular multiplicative SDEs via regularization by noise

TL;DR

This paper proves the existence of weak solutions for certain singular multiplicative stochastic differential equations driven by fractional Brownian motion, under minimal regularity conditions on the diffusion coefficient.

Contribution

It introduces a novel approach to establish weak solutions for singular SDEs using regularization by noise with fractional Brownian motion.

Findings

Existence of weak solutions under minimal integrability conditions.

Applicable to singular diffusion coefficients.

Utilizes fractional Brownian motion with specific Hurst parameters.

Abstract

We study multiplicative SDEs perturbed by an additive fractional Brownian motion on another probability space. Provided the Hurst parameter is chosen in a specified regime, we establish existence of probabilistically weak solutions to the SDE if the measurable diffusion coefficient merely satisfies an integrability condition. In particular, this allows to consider certain singular diffusion coefficients.