Estimation of the global regularity of a multifractional Brownian motion

TL;DR

This paper introduces a novel estimator for the global regularity of multifractional Brownian motion, utilizing a ratio statistic based on quadratic variations at different frequencies, with proven convergence properties.

Contribution

It proposes a new ratio-based estimator for the global regularity index of multifractional Brownian motion, demonstrating its convergence under weak assumptions.

Findings

The estimator converges in probability to the global regularity index.

The method effectively compares quadratic variations at two frequencies.

It provides a practical approach for estimating regularity in multifractional processes.

Abstract

This paper presents a new estimator of the global regularity index of a multifractional Brownian motion. Our estimation method is based upon a ratio statistic, which compares the realized global quadratic variation of a multifractional Brownian motion at two different frequencies. We show that a logarithmic transformation of this statistic converges in probability to the minimum of the Hurst function, which is, under weak assumptions, identical to the global regularity index of the path.