Parameter estimation for an Ornstein-Uhlenbeck 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 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-Esséen 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:
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arXiv e-prints
- Pub Date:
- November 2021
- DOI:
- 10.48550/arXiv.2111.15292
- arXiv:
- arXiv:2111.15292
- Bibcode:
- 2021arXiv211115292C
- Keywords:
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- Mathematics - Statistics Theory;
- Mathematics - Probability;
- 60H07;
- 60F25;
- 62M09
- E-Print:
- 35 pages. arXiv admin note: text overlap with arXiv:2002.09641