Parameter estimation for an Ornstein-Uhlenbeck Process driven by a general Gaussian noise with Hurst Parameter $H\in (0,\frac12)$

TL;DR

This paper studies parameter estimation for an Ornstein-Uhlenbeck process driven by a Gaussian noise with Hurst parameter H in (0, 1/2), establishing consistency and asymptotic normality of estimators using new Hilbert space relationships.

Contribution

It extends previous work to the case H in (0, 1/2), providing new theoretical results on estimator properties and establishing a novel relationship between associated Hilbert spaces.

Findings

Proves strong consistency of estimators for H in (0, 1/2)

Establishes asymptotic normality and Berry-Esséen bounds for H in (0, 3/8)

Develops new Hilbert space relationships for Gaussian processes with H in (0, 1/2)

Abstract

In Chen and Zhou 2021, they consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other is bounded by $(ts)^{H-1}$ with $H\in (\frac12,\,1)$, up to a constant factor. In this paper, we investigate the same problem but with the assumption of $H\in (0,\,\frac12)$. It is well known that there is a significant difference between the Hilbert space associated with the fractional Gaussian processes in the case of $H\in (\frac12, 1)$ and that of $H\in (0, \frac12)$. The starting point of this paper is a new relationship between the inner product of $\mathfrak{H}$ associated with the Gaussian process $(G_t)_{t\ge 0}$ and that of the Hilbert space $\mathfrak{H}_1$ associated with the fractional Brownian motion $(B^{H}_t)_{t\ge 0}$. Then we prove the strong consistency with $H\in (0, \frac12)$, and the asymptotic normality and the Berry-Ess\'{e}en bounds with $H\in (0,\frac38)$ for both the least squares estimator and the moment estimator of the drift parameter constructed from the continuous observations. A good many inequality estimates are involved in and we also make use of the estimation of the inner product based on the results of $\mathfrak{H}_1$ in Hu, Nualart and Zhou 2019.