Optimal estimation of the rough Hurst parameter in additive noise

TL;DR

This paper investigates the optimal estimation of the Hurst parameter in fractional Brownian motion from noisy high-frequency data, establishing minimax rates and constructing efficient estimators under various noise regimes.

Contribution

It introduces a new estimation procedure that achieves optimal convergence rates for the Hurst parameter in the presence of additive noise, extending classical results.

Findings

Establishes LAN property with rate n^{-1/2} when noise is small.

Derives minimax convergence rate (n/τ_n^2)^{-1/(4H+2)} for larger noise.

Provides a central limit theorem with explicit variance for the estimator.

Abstract

We estimate the Hurst parameter $H \in (0,1)$ of a fractional Brownian motion from discrete noisy data, observed along a high frequency sampling scheme. When the intensity $\tau_n$ of the noise is smaller in order than $n^{-H}$ we establish the LAN property with optimal rate $n^{-1/2}$. Otherwise, we establish that the minimax rate of convergence is $(n/\tau_n^2)^{-1/(4H+2)}$ even when $\tau_n$ is of order 1. Our construction of an optimal procedure relies on a Whittle type construction possibly pre-averaged, together with techniques developed in Fukasawa et al. [Is volatility rough? arXiv:1905.04852, 2019]. We establish in all cases a central limit theorem with explicit variance, extending the classical results of Gloter and Hoffmann [Estimation of the Hurst parameter from discrete noisy data. The Annals of Statistics, 35(5):1947-1974, 2007].