Strong uniqueness for fractional stochastic differential equations with sign drift

Abstract

Establishing pathwise uniqueness for stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>1/2$ remains challenging, particularly for discontinuous drifts beyond the regularity regimes of existing theories. A canonical example is the sign drift, for which uniqueness fails in the corresponding deterministic equation. In this paper, we complement recent strong existence results by establishing pathwise uniqueness for the equation with sign drift perturbed by fractional Brownian motion for every $H\in(1/2,1)$, thus obtaining strong well-posedness. Our proof develops a new approach based on the infinite iteration of an inequality arising from a fixed-point relation, combined with tail probability estimates exploiting the Gaussian structure of the noise. The methods and results cover a broader class of Gaussian Volterra processes satisfying suitable local nondeterminism and positivity conditions.