On ergodic least-squares estimators of the generalized diffusion coefficient for fractional Brownian motion

TL;DR

This paper introduces ergodic least-squares estimators for the generalized diffusion coefficient of fractional Brownian motion, demonstrating their convergence properties and optimal performance near a specific Hurst index value.

Contribution

It proposes a new class of ergodic estimators based on weighted functionals of single trajectories, improving diffusion coefficient estimation from limited data.

Findings

Estimators are ergodic and converge to the true diffusion coefficient.

Convergence is fastest around H≈0.30, in the subdiffusive regime.

Higher experimental resolution enhances estimation accuracy.

Abstract

We analyse a class of estimators of the generalized diffusion coefficient for fractional Brownian motion $B_t$ of known Hurst index $H$, based on weighted functionals of the single time square displacement. We show that for a certain choice of the weight function these functionals possess an ergodic property and thus provide the true, ensemble-averaged, generalized diffusion coefficient to any necessary precision from a single trajectory data, but at expense of a progressively higher experimental resolution. Convergence is fastest around $H\simeq0.30$, a value in the subdiffusive regime.