Estimation of the Hurst parameter from continuous noisy data

TL;DR

This paper investigates how to accurately estimate the Hurst exponent of fractional Brownian motion from noisy continuous data, establishing the conditions for consistent estimation and deriving optimal rates through LAN analysis.

Contribution

It proves the Local Asymptotic Normality of the model under increasing observation length or decreasing noise, revealing the optimal minimax rates for estimation.

Findings

LAN property established for the model

Optimal minimax rates identified

Consistent estimation possible under specified regimes

Abstract

This paper addresses the problem of estimating the Hurst exponent of the fractional Brownian motion from continuous time noisy sample. Consistent estimation in the setup under consideration 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 rates.