Fast convergence rates for estimating the stationary density in SDEs driven by a fractional Brownian motion with semi-contractive drift

We consider the solution of an additive fractional stochastic differential equation (SDE) and, leveraging continuous observations of the process, introduce a methodology for estimating its stationary density $\pi$. Initially, employing a tailored martingale decomposition specifically designed for the statistical challenge at hand, we establish convergence rates surpassing those found in existing literature. Subsequently, we refine the attained rate for the case where $H < \frac{1}{2}$ by incorporating bounds on the density of the semi-group. This enhancement outperforms previous rates. Finally, our results weaken the usual convexity assumptions on the drift component, allowing to consider settings where strong convexity only holds outside a compact set.