Statistical analysis of the non-ergodic fractional Ornstein-Uhlenbeck process with periodic mean

08/31/2020

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by   Rachid Belfadli, et al.

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Consider a periodic, mean-reverting Ornstein-Uhlenbeck process X={X_t,t≥0} of the form d X_t=(L(t)+α X_t) d t+ dB^H_t, t ≥ 0, where L(t)=∑_i=1^pμ_iϕ_i (t) is a periodic parametric function, and {B^H_t,t≥0} is a fractional Brownian motion of Hurst parameter 1/2≤ H<1. In the "ergodic" case α<0, the parametric estimation of (μ_1,…,μ_p,α) based on continuous-time observation of X has been considered in Dehling et al. <cit.>, and in Dehling et al. <cit.> for H=1/2, and 1/2<H<1, respectively. In this paper we consider the "non-ergodic" case α>0, and for all 1/2≤ H<1. We analyze the strong consistency and the asymptotic distribution for the estimator of (μ_1,…,μ_p,α) when the whole trajectory of X is observed.