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

TL;DR

This paper introduces a kernel-based estimator for the invariant density of fractional SDEs that achieves faster convergence rates, extends to discrete data, and adapts to unknown smoothness, even under weakened convexity conditions.

Contribution

It develops a new estimator with improved convergence rates for fractional SDEs, including discrete observations and relaxed drift assumptions, with practical adaptive procedures.

Findings

Faster convergence rates for invariant density estimation in fractional SDEs.

Effective extension of the estimator to discrete observation data.

Numerical validation demonstrating practical performance.

Abstract

We study the estimation of the invariant density of additive fractional stochastic differential equations with Hurst parameter $H \in (0,1)$. We first focus on continuous observations and develop a kernel-based estimator achieving faster convergence rates than previously available. This result stems from a martingale decomposition combined with new bounds on the (conditional) convergence in total variation to equilibrium of fractional SDEs. For $H<1/2$, we further refine the rates based on recent bounds on the marginal density. We then extend the methodology to discrete observations, showing that the same convergence rates can be attained. Moreover, we establish concentration inequalities for the estimator and introduce a data-driven bandwidth selection procedure that adapts to unknown smoothness. Numerical experiments for the fractional Ornstein-Uhlenbeck process illustrate the estimator's practical performance. Finally, our results weaken the usual convexity assumptions on the drift component, allowing us to consider settings where strong convexity only holds outside a compact set.