Estimation of the Hurst parameter from continuous noisy data

Abstract

This paper addresses the problem of estimating the Hurst exponent of the fractional Brownian motion from continuous time noisy sample. When the Hurst parameter is greater than $3∕4$, consistent estimation is possible only if either the length of the observation interval increases to infinity or intensity of the noise decreases to zero. The main result is a proof of the Local Asymptotic Normality (LAN) of the model in these two regimes which reveals the optimal minimax estimation rates.