  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.
