Optimal estimation of the rough Hurst parameter in additive noise

Grégoire Szymanski

Stochastic Processes and their Applications, 2024, vol. 170, issue C

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).

Keywords: Scaling exponent; high-frequency data; Fractional Brownian motion; Whittle estimator; LAN property (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spa.2024.104302

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