Parameter Estimation for an Ornstein-Uhlenbeck Process Driven by a General Gaussian Noise

Abstract

In this paper, we 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 $R(t,s) = \mathbb{E}\left[ {{G_t}{G_s}} \right]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by (ts)^β−1 up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or to have stationary increments; some examples of this include the subfractional Brownian motion and the bi-fractional Brownian motion. Under this assumption, we study the parameter estimation for a drift parameter in the Ornstein-Uhlenbeck process driven by the Gaussian noise (G_t)_t≥0. For the least squares estimator and the second moment estimator constructed from the continuous observations, we prove the strong consistency and the asympotic normality, and obtain the Berry-Esséen bounds. The proof is based on the inner product’s representation of the Hilbert space $\mathfrak{h}$ associated with the Gaussian noise (G_t)_t≥0, and the estimation of the inner product based on the results of the Hilbert space associated with the fractional Brownian motion.