Least squares estimator of fractional Ornstein–Uhlenbeck processes with periodic mean for general Hurst parameter

Abstract

In this paper, we deal with least squares estimator for the drift parameters of the fractional Ornstein–Uhlenbeck process with periodic mean function for all the Hurst parameter range $H\in (0,1)$. More precisely, we extend the strong consistency proved in Bajja et al. (J Korean Stat Soc 46:608–622, 2017) for $\frac{1}{2}<H<1$ to more general Hurst parameter $0<H<1$. We also discuss the asymptotic normality given in Dehling et al. (Stat Inference Stoch Process 20:1–14, 2017) and Bajja et al. (J Korean Stat Soc 46:608–622, 2017). By the results in Hu et al. (Stat Inference Stoch Process 22:111–142, 2019), two different central limit theorems are proved when the Hurst parameter $H\in (0,\frac{1}{2})$ and $H\in (\frac{1}{2},1)$, respectively.

Appendix

In this section, we turn to the Lemmas give the convergence of the stochastic integral driven by fBm are available in Lemmas 17-19 of Hu et al. (2019) and Hu and Nualart (2010), which technical results used in various proofs of this paper.

Lemma 6.1

Let

$$\begin{aligned} F_T=\int _0^T\int _0^Te^{-\alpha |t-s|}dB_t^HdB_s^H \end{aligned}$$

and

$$\begin{aligned} \widetilde{F}_T=T^{2H}\int _0^1\int _0^1e^{-\alpha T|t-s|}dB_t^HdB_s^H. \end{aligned}$$

Moreover, let $S_1=I_2(\delta _{0,1})$ is the Rosenblatt random variable and $\delta _{0,1}$ is the Dirac-type distributed random variable defined in (2.6) of Hu et al. (2019). Then we have the following convergence results.

• (i) When $0<H<\frac{1}{2}$, we have

  $$\begin{aligned} \lim _{T\rightarrow \infty }\mathbb {E}(\frac{1}{T}F^2_T)=4H^2\alpha ^{1-4H}(\varGamma (2H))^2 \Big ((4H-1)+\frac{2\varGamma (2-4H)\varGamma (4H)}{\varGamma (1-2H)\varGamma (2H)}\Big ). \end{aligned}$$

• (ii) When $\frac{1}{2}\le H<\frac{3}{4}$, we have

  $$\begin{aligned} \lim _{T\rightarrow \infty }\mathbb {E}(\frac{1}{T}F^2_T)=4H^2(4H-1)\alpha ^{1-4H} \Big ((\varGamma (2H))^2+\frac{\varGamma (2H)\varGamma (3-4H)\varGamma (4H-1)}{\varGamma (2-2H)}\Big ). \end{aligned}$$

• (iii) When $H=\frac{3}{4}$, we have

  $$\begin{aligned} \lim _{T\rightarrow \infty }\frac{\mathbb {E}(F^2_T)}{T\log (T)}=\frac{9}{4}\alpha ^{-2}. \end{aligned}$$

• (iv) When $\frac{3}{4}<H<1$, we have

  $$\begin{aligned} \lim _{T\rightarrow \infty }\mathbb {E}(T^{2-4H}F^2_T)=\frac{16\beta _H^2\alpha ^{-2}}{(4H-2)(4H-3)} \end{aligned}$$

  and

  $$\begin{aligned} \lim _{T\rightarrow \infty }\mathbb {E}(T^{1-2H}S_1\widetilde{F}_T)=\frac{8\beta _H^2 \alpha ^{-1}}{(4H-2)(4H-3)}, \end{aligned}$$

  where $\beta _H=H(2H-1)$.

Lemma 6.2

Let the stochastic process $Y_t=e^{-\alpha t}\int _0^te^{\alpha s}dB_s^H$. Then, as $T\rightarrow \infty$,

$$\begin{aligned} \frac{1}{T}\int _0^TY_t^2dt\rightarrow \alpha ^{-2H}H\varGamma (2H), ~~a.s. \text {and in} ~~L^2. \end{aligned}$$

(6.1)

Lemma 6.3

Let $\widetilde{Y}_t$ be defined by

$$\begin{aligned} \widetilde{Y}_t=e^{-\alpha t}\int _{-\infty }^te^{\alpha s}dB_s^H=Y_t+e^{-\alpha t}\xi , \end{aligned}$$

where $\xi =\int _{-\infty }^0e^{\alpha s}dB_s^H$. Then, $\widetilde{Y}_t$ is Gaussian, stationary and ergodic for all $H\in (0,1)$ and for any $\beta >0$, $\frac{\widetilde{Y}_T}{T^{\beta }}$ converges almost surely to zero as T tends to infinity.