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

Abstract

Consider a periodic, mean-reverting Ornstein–Uhlenbeck process \(X=\{X_t,t\ge 0\}\) of the form \(d X_{t}=\left( L(t)+\alpha X_{t}\right) d t+ dB^H_{t}, \quad t \ge 0\), where \(L(t)=\sum _{i=1}^{p}\mu _i\phi _i (t)\) is a periodic parametric function, and \(\{B^H_t,t\ge 0\}\) is a fractional Brownian motion of Hurst parameter \(\frac{1}{2}\le H<1\). In the “ergodic” case \(\alpha <0\), the parametric estimation of \((\mu _1,\ldots ,\mu _p,\alpha )\) based on continuous-time observation of X has been considered in Dehling et al. (Stat Inference Stoch Process 13:175–192, 2010; Stat Inference Stoch Process 20:1–14, 2016) for \(H=\frac{1}{2}\), and \(\frac{1}{2}<H<1\), respectively. In this paper we consider the “non-ergodic” case \(\alpha >0\), and for all \(\frac{1}{2}\le H<1\). We analyze the strong consistency and the asymptotic distribution for the estimator of \((\mu _1,\ldots ,\mu _p,\alpha )\) when the whole trajectory of X is observed.