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

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 $H\in (0,\,\frac12)$. The starting point of this paper is a new relationship between the inner product of $\mathfrak{H}$ and that of the Hilbert space $\mathfrak{H}_1$ associated with the fractional Brownian motion $(B^{H}_t)_{t\ge 0}$. Based on this relationship and some known estimation of the inner product of $\mathfrak{H}_1$, 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.