Parameter estimation for an Ornstein-Uhlenbeck Process driven by a general Gaussian noise with Hurst Parameter H∈ (0,1/2)
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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≥ 0. The second order mixed partial derivative of the covariance function R(t, s)=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∈ (1/2, 1), up to a constant factor. In this paper, we investigate the same problem but with the assumption of H∈ (0, 1/2). 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∈ (1/2, 1) and that of H∈ (0, 1/2). The starting point of this paper is a new relationship between the inner product of ℌ associated with the Gaussian process (G_t)_t≥ 0 and that of the Hilbert space ℌ_1 associated with the fractional Brownian motion (B^H_t)_t≥ 0. Then we prove the strong consistency with H∈ (0, 1/2), and the asymptotic normality and the Berry-Esséen bounds with H∈ (0,3/8) 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 ℌ_1 in Hu, Nualart and Zhou 2019.
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