Optimal estimation of the rough Hurst parameter in additive noise

Abstract

We estimate the Hurst parameter H∈(0,1) of a fractional Brownian motion from discrete noisy data, observed along a high-frequency sampling scheme. When the intensity τn 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/τn2)−1/(4H+2) even when τn 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. (2019). We establish in all cases a central limit theorem with explicit variance, extending the classical results of Gloter and Hoffmann (2007).