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.

考虑一个周期性的均值回复 Ornstein–Uhlenbeck 过程\(X=\{X_t,t\ge 0\}\)，其形式为\(d X_{t}=\left( L(t)+\alpha X_{ t}\right) d t+ dB^H_{t}, \quad t \ge 0\)，其中\(L(t)=\sum _{i=1}^{p}\mu _i\phi _i ( t)\)是周期参数函数，\(\{B^H_t,t\ge 0\}\)是赫斯特参数的分数布朗运动\(\frac{1}{2}\le H<1 \)。在“遍历”情况下\(\alpha <0\) ，基于X的连续时间观察的\((\mu _1,\ldots ,\mu _p,\alpha )\)的参数估计已在德林等人。(Stat Inference Stoch Process 13:175–192, 2010;Stat Inference Stoch Process 20:1–14, 2016)\(H=\frac{1}{2}\)和\(\frac{1}{2}<H<1\)分别。在本文中，我们考虑“非遍历”情况\(\alpha >0\)，并且对于所有\(\frac{1}{2}\le H<1\)。当观察到X的整个轨迹时，我们分析了\((\mu _1,\ldots ,\mu _p,\alpha )\)的估计量的强一致性和渐近分布。