Estimation of the Hurst parameter from discrete noisy data

TL;DR

This paper develops optimal methods for estimating the Hurst parameter of fractional Brownian motion from high-frequency noisy data, addressing the challenge posed by experimental noise affecting high-frequency information.

Contribution

It introduces rate-optimal estimators for the Hurst parameter that adaptively estimate quadratic functionals, establishing the minimax optimal rate of convergence.

Findings

Proves the minimax optimal rate of $n^{-1/(4H+2)}$ for estimation.

Develops adaptive estimators based on quadratic functionals.

Addresses the impact of high-frequency noise on parameter estimation.

Abstract

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.