Parameter estimation for an OrnsteinUhlenbeck Process driven by a general Gaussian noise with Hurst Parameter $H\in (0,\frac12)$
Abstract
In Chen and Zhou 2021, they consider an inference problem for an OrnsteinUhlenbeck process driven by a general onedimensional 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 $(ts)^{H1}$ 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 BerryEsséen 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.15292
 Bibcode:
 2021arXiv211115292C
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Probability;
 60H07;
 60F25;
 62M09
 EPrint:
 35 pages. arXiv admin note: text overlap with arXiv:2002.09641