Optimal estimation of the rough Hurst parameter in additive noise
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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 τ_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/τ_n^2)^-1/(4H+2) even when τ_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].
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