Optimal estimation of the rough Hurst parameter in additive noise

Understand

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.