Quasi-Likelihood Analysis of Fractional Brownian Motion with Constant Drift under High-Frequency Observations

TL;DR

This paper introduces a consistent and asymptotically normal estimator for the Hurst parameter and volatility of fractional Brownian motion with constant drift, based on high-frequency data and quasi-likelihood methods.

Contribution

It proposes a novel estimator combining local Gaussian approximation and frequency-domain methods for fractional Brownian motion with constant drift.

Findings

Estimator is consistent for all H in (0,1)

Estimator is asymptotically normal under high-frequency observations

Applicable to fractional Brownian motion with constant drift

Abstract

Consider an estimation of the Hurst parameter $H\in(0,1)$ and the volatility parameter $\sigma>0$ for a fractional Brownian motion with a drift term under high-frequency observations with a finite time interval. In the present paper, we propose a consistent estimator of the parameter $\theta=(H,\sigma)$ combining the ideas of a quasi-likelihood function based on a local Gaussian approximation of a high-frequently observed time series and its frequency-domain approximation. Moreover, we prove an asymptotic normality property of the proposed estimator for all $H\in(0,1)$ when the drift process is constant.