Strong solutions for singular SDEs driven by long-range dependent fractional Brownian motion and other Volterra processes

Abstract

We investigate the well-posedness of stochastic differential equations driven by fractional Brownian motion, focusing on the long-range dependent case $H\in (\frac{1}{2},1)$. While existing results on regularization by such noise typically require Hölder continuity of the drift, we establish new strong existence and uniqueness results for certain classes of singular drifts, including discontinuous and highly irregular functions. More generally, we treat stochastic differential equations with additive noise given by a broader class of Volterra processes satisfying suitable kernel conditions, which, in addition to fractional Brownian motion, also includes the Riemann-Liouville process as a special case. Our approach relies on probabilistic arguments.