Strong solutions of SDE's with generalized drift and multidimensional fractional Brownian initial noise

TL;DR

This paper establishes the existence of strong solutions for multidimensional fractional Brownian motion-driven SDEs with generalized drift defined via local time, extending skew Brownian motion concepts to fractional cases.

Contribution

It introduces a novel approach combining Malliavin calculus and local time variational calculus to prove strong solutions for SDEs with generalized drift driven by fractional Brownian motion.

Findings

Proves existence of strong solutions for H<1/2.

Extends skew Brownian motion to fractional Brownian context.

Develops new techniques combining Malliavin calculus and local time methods.

Abstract

In this paper we prove the existence of strong solutions to a SDE with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<1/2. Here the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a "local time variational calculus" argument.