Parameter estimation of non-ergodic Ornstein-Uhlenbeck
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In this paper, we consider the statistical inference of the drift parameter θ of non-ergodic Ornstein-Uhlenbeck (O-U) process driven by a general Gaussian process (G_t)_t≥ 0. When H ∈ (0, 1/2) ∪ (1/2,1) the second order mixed partial derivative of R (t, s) = E [G_t G_s] can be decomposed into two parts, one of which coincides with that of fractional Brownian motion (fBm), and the other of which is bounded by |ts|^H-1. This condition covers a large number of common Gaussian processes such as fBm, sub-fractional Brownian motion and bi-fractional Brownian motion. Under this condition, we verify that (G_t)_t≥ 0 satisfies the four assumptions in references <cit.>, that is, noise has Hölder continuous path; the variance of noise is bounded by the power function; the asymptotic variance of the solution X_T in the case of ergodic O-U process X exists and strictly positive as T →∞; for fixed s ∈ [0,T), the noise G_s is asymptotically independent of the ergodic solution X_T as T →∞, thus ensure the strong consistency and the asymptotic distribution of the estimator θ̃_T based on continuous observations of X. Verify that (G_t)_t≥ 0 satisfies the assumption in references <cit.>, that is, the variance of the increment process {ζ_t_i-ζ_t_i -1, i =1,..., n } is bounded by the product of a power function and a negative exponential function, which ensure that θ̂_n and θ̌_n are strong consistent and the sequences √(T_n) (θ̂_n - θ) and √(T_n) (θ̌_n - θ) are tight based on discrete observations of X
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