Statistical analysis of the non-ergodic fractional Ornstein-Uhlenbeck process with periodic mean

TL;DR

This paper investigates the statistical properties of estimators for a non-ergodic fractional Ornstein-Uhlenbeck process with periodic mean, focusing on strong consistency and asymptotic distribution for all Hurst parameters in the range [0.5, 1).

Contribution

It extends previous work by analyzing the non-ergodic case with positive mean reversion parameter and fractional Brownian motion, providing new results on estimator consistency and distribution.

Findings

Establishes strong consistency of estimators in the non-ergodic case.

Derives the asymptotic distribution of estimators for all H in [0.5, 1).

Analyzes the impact of periodic mean and fractional noise on estimation.

Abstract

Consider a periodic, mean-reverting Ornstein-Uhlenbeck process $X=\{X_t,t\geq0\}$ of the form $d X_{t}=\left(L(t)+\alpha X_{t}\right) d t+ dB^H_{t}, \quad t \geq 0$, where $L(t)=\sum_{i=1}^{p}\mu_i\phi_i (t)$ is a periodic parametric function, and $\{B^H_t,t\geq0\}$ is a fractional Brownian motion of Hurst parameter $\frac12\leq H<1$. In the "ergodic" case $\alpha<0$, the parametric estimation of $(\mu_1,\ldots,\mu_p,\alpha)$ based on continuous-time observation of $X$ has been considered in Dehling et al. \cite{DFK}, and in Dehling et al. \cite{DFW} for $H=\frac12$, and $\frac12<H<1$, respectively. In this paper we consider the "non-ergodic" case $\alpha>0$, and for all $\frac12\leq H<1$. We analyze the strong consistency and the asymptotic distribution for the estimator of $(\mu_1,\ldots,\mu_p,\alpha)$ when the whole trajectory of $X$ is observed.