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

TL;DR

The paper tackles nonparametric estimation of the stationary (invariant) density for additive SDEs driven by fractional Brownian motion with Hurst parameter $H$. It introduces a kernel-based estimator and, through a novel martingale decomposition and total variation contraction bounds, establishes faster convergence rates than prior work, with rates depending on the Hölder smoothness and $H$. The results extend from continuous to discrete observations and include a data-driven bandwidth selection method that achieves the same rates adaptively, even under weaker semi-contractive drift assumptions. Numerical experiments on fractional Ornstein–Uhlenbeck models illustrate practical performance, confirming the theoretical rates, while the framework allows the drift to be strongly contractive only outside a compact set, broadening applicability.

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.

Figures (2)

• Figure 1: Comparison between true and estimated densities. Left: $H=0.3$ with $h$ of Theo 1 and Theo 2. Right $H=0.7$.
• Figure 2: Evolution of the MSE in logarithmic scale. Blue line: theoretical exponent of Th. \ref{['th: rate start discrete']}, dotted lines: MSE. Top: Gaussian kernel ($M=2$). Bottom: Legendre kernel with $M=5$.

Theorems & Definitions (46)

• Proposition 1
• Remark 1
• Definition 1
• Remark 2
• Proposition 2
• Theorem 1
• Remark 3
• Remark 4
• Remark 5
• Proposition 3