Consistency of the drift parameter estimator for the discretized fractional Ornstein-Uhlenbeck process with Hurst index $H\in(0,\frac12)$

TL;DR

This paper studies the consistency of a drift parameter estimator for a fractional Ornstein-Uhlenbeck process driven by fractional Brownian motion with Hurst index less than 0.5, using high-frequency discrete observations.

Contribution

It introduces a new estimator for the drift parameter in fractional Ornstein-Uhlenbeck processes and proves its strong consistency under high-frequency sampling.

Findings

Estimator is strongly consistent for positive drift for any m>1.

Estimator is consistent for negative drift when m>1/(2H).

Results apply to high-frequency data with decreasing observation intervals.

Abstract

We consider Langevin equation involving fractional Brownian motion with Hurst index $H\in(0,\frac12)$. Its solution is the fractional Ornstein-Uhlenbeck process and with unknown drift parameter $\theta$. We construct the estimator that is similar in form to maximum likelihood estimator for Langevin equation with standard Brownian motion. Observations are discrete in time. It is assumed that the interval between observations is $n^{-1}$, i.e. tends to zero (high frequency data) and the number of observations increases to infinity as $n^m$ with $m>1$. It is proved that for positive $\theta$ the estimator is strongly consistent for any $m>1$ and for negative $\theta$ it is consistent when $m>\frac{1}{2H}$.