# Inference for fractional Ornstein-Uhlenbeck type processes with periodic mean in the non-ergodic case

## TL;DR

This paper develops estimators for the drift parameters of a fractional Ornstein-Uhlenbeck process with periodic mean, establishing their strong consistency and asymptotic distributions in the non-ergodic, long-range dependent case.

## Contribution

It introduces a least-squares estimator for the non-ergodic fractional Ornstein-Uhlenbeck process with periodic mean and proves its strong consistency and asymptotic normality.

## Key findings

- Estimator is strongly consistent for known Hurst parameter.
- Asymptotic normality with rate depending on Hurst parameter.
- Improved rate of convergence for certain periodic functions.

## Abstract

In the paper we consider the problem of estimating parameters entering the drift of a fractional Ornstein-Uhlenbeck type process in the non-ergodic case, when the underlying stochastic integral is of Young type. We consider the sampling scheme that the process is observed continuously on $[0,T]$ and $T\to\infty$. For known Hurst parameter $H\in(0.5, 1)$, i.e. the long range dependent case, we construct a least-squares type estimator and establish strong consistency. Furthermore, we prove a second order limit theorem which provides asymptotic normality for the parameters of the periodic function with a rate depending on $H$ and a non-central Cauchy limit result for the mean reverting parameter with exponential rate. For the special case that the periodicity parameter is the weight of a periodic function, which integrates to zero over the period, we can even improve the rate to $\sqrt{T}$.