Estimation of the Hurst parameter from discrete noisy data

We estimate the Hurst parameter H of a fractional Brownian motion from discrete noisy data observed along a high frequency sampling scheme. The presence of systematic experimental noise makes recovery of H more difficult since relevant information is mostly contained in the high frequencies of the signal. We quantify the difficulty of the statistical problem in a min-max sense: we prove that the rate n(-1/(4H+2)) is optimal for estimating H and propose rate optimal estimators based on adaptive estimation of quadratic functionals.