Confidence intervals for the Hurst parameter of a fractional Brownian motion based on finite sample size

TL;DR

This paper develops exact confidence intervals for the Hurst parameter of fractional Brownian motion using concentration inequalities, applicable with finite samples and without prior parameter assumptions.

Contribution

It introduces a novel method leveraging Gaussian quadratic form concentration inequalities to construct confidence intervals for the Hurst parameter, handling known and unknown scale cases.

Findings

Provides exact confidence intervals based on finite samples

Applicable to both known and unknown scale parameters

Does not require assumptions on the Hurst parameter

Abstract

In this paper, we show how concentration inequalities for Gaussian quadratic form can be used to propose exact confidence intervals of the Hurst index parametrizing a fractional Brownian motion. Both cases where the scaling parameter of the fractional Brownian motion is known or unknown are investigated. These intervals are obtained by observing a single discretized sample path of a fractional Brownian motion and without any assumption on the parameter $H$.