Abstract:

In 2021, Chen and Zhou consider an inference problem for an Ornstein-Uhlenbeck process driven by a type of centered fractional 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 quantitative relation 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 $(B^{H}_t)_{t\ge 0}$. 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 based on the continuous observations.

Key words: Fractional Brownian motion, Fourth moment theorems, Ornstein-Uhlenbeck process, Fractional Gaussian process, Berry-Esséen type upper bounds