Distribution dependent SDEs driven by additive fractional Brownian motion

TL;DR

This paper investigates distribution dependent stochastic differential equations driven by fractional Brownian motion with irregular drifts, establishing strong well-posedness under broad conditions and extending prior results.

Contribution

It extends existing results on SDEs driven by fractional Brownian motion to the distribution dependent case with irregular drifts, using novel stability estimates and Wasserstein distances.

Findings

Established strong well-posedness for distribution dependent SDEs with irregular drifts.

Extended previous results to include distribution dependent cases with fractional Brownian motion.

Developed new stability estimates for singular SDEs driven by fractional Brownian motion.

Abstract

We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $H\in (0,1)$. We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$B(\cdot,\mu) = f\ast\mu(\cdot) + g(\cdot),\quad f,g\in B^\alpha_{\infty,\infty}, \quad \alpha>1-1/2H,$$ thus extending the results by Catellier and Gubinelli [9] to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances.