Least squares type estimations for discretely observed nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind

1.   Introduction and main results

Recently, the study of fractional Ornstein-Uhlenbeck processes of the second kind (FOUSK) has attracted interest. For example, Azmoodeh and Morlanes (2013) studied the drift parameter estimation of FOUSK based on continuous observations in the ergodic case. Azmoodeh and Viitasaari (2015) considered the drift parameter estimation of FOUSK based on discrete observations in the ergodic case. For the nonergodic case, the drift parameter estimations of FOUSK based on continuous and discrete observations were studied in EI Onsy et al. (2017) and EI Onsy et al. (2018), respectively. Balde et al. (2018) investigated the infinite-dimensional version of FOUSK. Yu et al. (2017) studied the problem of parameter estimation for Ornstein-Uhlenbeck processes of the second kind driven by $\alpha$-stable Lévy motions, based on continuous and discrete observations, respectively. Es-Sebaiy et al. (2019) considered least squares type estimations for discretely observed nonergodic Gaussian Ornstein-Uhlenbeck processes.

Motivated by all these studies, in this paper, we will consider the nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind (GOUSK) defined by

with an unknown parameter $\theta > 0,$ where $dY_t^{(1)} = e^{-t}dG_{a_{t}}$ and $G = \{G_t, t\geq 0\}$ is a mean zero Gaussian process with the self-similar index $\gamma\in (\frac{1}{2}, 1)$ and $a_t = \gamma e^{\frac{t}{\gamma}}$. Based on the discrete observations $\{X_{t_i}:t_i = i\Delta_n, i = 0, 1, \cdots, n\}$, we construct two least squares type estimators $\hat{\theta}_n$ and $\tilde{\theta}_n$ of $\theta$:

and

where $T_n = n\Delta_n$ denotes the length of the 'observation window'.

Let the covariance function of Gaussian process $G$ be $R(t, s) = {\bf E}(G_tG_s), t\geq 0, s\geq 0.$ Denote

Assume that

where $c(\gamma)$ is a positive constant depending on $\gamma$, and $\frac{1}{2} < \gamma < 1$ is self-similar index of $G$.

For the assumption (1.5), many self-similar Gaussian processes satisfy the condition, such as fractional Brownian motion, sub-fractional Brownian motion, bifractional Brownian motion and sub-bifractional Brownian motion. Therefore our main results hold for the Gaussian processes mentioned above.

Let $\{Z_n\}$ be a sequence of random variables. We say $\{Z_n\}$ is tight (or bounded in probability) if for every $\epsilon > 0$, there exists $M_{\epsilon} > 0,$ such that

Now we state our main results as follows.

Theorem 1.1. Assume that (1.5) holds and $\frac{1}{2} < \gamma < 1$, and $\theta > 0, \Delta_n\rightarrow 0$ and $n\Delta_n^{1+\beta}\rightarrow \infty$ for some $\beta > 0$ as $n\rightarrow \infty$. Then, we have, as $n\rightarrow \infty,$

and

Theorem 1.2. Assume that (1.5) holds and $\frac{1}{2} < \gamma < 1$, and $\theta > 0, \Delta_n\rightarrow 0$ and $n\Delta_n^{1+\beta}\rightarrow \infty$ for some $\beta > 0$ as $n\rightarrow \infty$. Then, we have

(1) for any $q\geq 0,$

(2) if $n\Delta_n^3\rightarrow 0$ as $n\rightarrow \infty$, then

Theorem 1.3. Assume that (1.5) holds and $\frac{1}{2} < \gamma < 1$, and $\theta > 0, \Delta_n\rightarrow 0$ and $n\Delta_n^{1+\beta}\rightarrow \infty$ for some $\beta > 0$ as $n\rightarrow \infty$. Then, we have

(1) for any $q\geq 0,$

(2) if $n\Delta_n^3\rightarrow 0$ as $n\rightarrow \infty$, then

Remark. We study Gaussian Ornstein-Uhlenbeck processes of the second kind in this paper. We know they are a subset of a much larger class Barndorff-Nielsen and Shephard type Ornstein-Uhlenbeck processes (see Barndorff-Nielsen (2001) and Barndorff-Nielsen and Shephard (2001)). In the future, we will extend our results to more general Ornstein-Uhlenbeck type models (see Salmon and SenGupta (2021), Issaka and SenGupta (2017) and Roberts and SenGupta (2020)).

We have organized our paper as follows: In Sect.2 we present some preliminaries for the Gaussian process $G$ and main lemmas. Sect.3 is devoted to the proofs of Theorems 1.1–1.3. In Sect.4 we apply our results to the cases such as fractional Brownian motion, sub-fractional Brownian motion, bifractional Brownian motion and sub-bifractional Brownian motion, while Sect.5 contains numerical simulations for four fractional Gaussian processes.

2.   Preliminaries and main lemmas

In this section, we firstly recall some elements of the Malliavin calculus. We refer to Nualart (2006) for detailed account these notions(see Moshrefi-Torbati and Hammond (1998), Meerschaert et al. (2017), Ei-Nabulsi (2012, 2015, 2017), and Ei-Nabulsi and Golmankhaneh (2021)). Let $\mathcal{H}$ be a real separable Hilbert space associated with the Gaussian process $G$, which is defined by the closure of the linear space $\varepsilon$ generated by the indicator functions $\{{\textbf I}_{[0, t]}, t\in [0, T]\}$ with respect to the scalar product

We know that the covariance of $G$ can be written as

where $\phi(u, v)$ is defined by (1.4).

We can find a linear space of functions contained in $\mathcal{H}$ in the following way. Let $|\mathcal{H}|$ be the linear space of measurable functions $\varphi$ on $[0, T]$ such that

It is not difficult to show that $|\mathcal{H}|$ is a Banach space with the norm $||\bullet||_{|\mathcal{H}|}$ and $\varepsilon$ is dense in $|\mathcal{H}|$.

Moreover, for all $\varphi, \psi\in |\mathcal{H}|,$ it can be proved that

For every $q\geq 1$, let $\mathcal{H}_q$ be the $q$th Wiener chaos of $G$, namely, the closed linear subspace of $L^2(\Omega)$ generated by the random variables $\{H_q(G(h)), h\in \mathcal{H}, ||h||_{\mathcal{H}} = 1\}$, where $H_q$ is the $q$th Hermite polynomial defined as $H_q(x) = (-1)^qe^{\frac{x^2}{2}}\frac{d^q}{dx^q}\left(e^{-\frac{x^2}{2}} \right)$. The mapping $I_q(h^{\otimes q}) = H_q(G(h))$ provides a linear isometry between the symmetric tensor product $\mathcal{H}^{\odot q}$ (equipped with the modified norm $||\cdot||_{\mathcal{H}^{\odot q}} = \sqrt{q!}||\cdot||_{\mathcal{H}^{\otimes q}}$) and $\mathcal{H}_q$. Specifically, for all $f, g\in \mathcal{H}^{\odot q}$ and $q\geq 1$, one has

For the multiple stochastic integral $I_q(f)$, it has the following property: for any $p\geq 2,$

where $c(p, q)$ is a positive constant only depending on $p$ and $q.$

It is easy to obtain the solution of (1.1):

where the integral with respect to $Y_s^{(1)}$ is a Young integral (see Young (1936)).

Denote

then,

By (2.6) and $dY_s^{(1)} = e^{-s}dG_{a_{s}}$, $a_s = \gamma e^{\frac{s}{\gamma}}$ and let $a_s = u$, we get

By (2.2) and $\frac{1}{2} < \gamma < 1$, we have, for $0\leq s < t$,

In order to prove Theorems 1.1-1.3, we need the following some lemmas.

Lemma 2.1. Let $\eta = \{\eta_t, t\geq 0\}$ be given by (2.6). Assume that (1.5) holds and $\frac{1}{2} < \gamma < 1$. Then,

(1) For all $\epsilon\in (0, \gamma)$, the process $\eta$ has a modification with $(\gamma-\epsilon)$-Hölder continuous paths, still denoted $\eta$ in the sequel.

(2) As $t\rightarrow \infty$,

almost surely and in $L^2(\Omega)$.

Proof. Since the proof is similar to that of Lemma 2.2 in EI Onsy et al. (2017), we omit the details.

Lemma 2.2. Let $l > 0$ and $\{Z_n\}_{n\in {\textbf N}}$ be a sequence of random variables. Suppose that, for every $p\geq 1,$ there exists a constant $c_p > 0$ such that, for all $n\in {\textbf N}$,

Then, for all $\epsilon > 0$, there exists a random variable $\xi_{\epsilon}$ such that, for any $n\in {\textbf N}$,

moreover, ${\textbf E}[|\xi_{\epsilon}|^p] < \infty$ for all $p\geq 1.$

Proof. See the proof in Kloeden and Neuenkirch (2007).

Lemma 2.3. Let

and

Then, we have

Assume that (1.5) holds and $\frac{1}{2} < \gamma < 1$, and $\theta > 0, \Delta_n\rightarrow 0$ and $n\Delta_n^{1+\beta}\rightarrow \infty$ for some $\beta > 0$ as $n\rightarrow \infty$, then,

In particular, as $n\rightarrow \infty,$ we have

where $\eta_{\infty}$ is defined by (2.10).

Proof. By (2.7), we obtain

which shows that (2.13) holds.

Using (2.9) and (1.5) and making the change of variables $r = \frac{u}{a_{t_{i-1}}}, \ s = \frac{v}{a_{t_{i-1}}}$, we get

By (2.10) and $\eta$ is Gaussian, we also obtain, for every $p\geq 1$,

where $c(p)$ is a constant depending on $p$. In fact, we have, by Cauchy-Schwarz inequality,

the last inequality comes from the fact: if $\xi$ is Gaussian and $p\geq 2$, then

By (2.17) and (2.16) and the Minkowski inequality, we deduce that

where $c$ is a constant, and the last inequality comes from the fact: as $x\rightarrow 0$,

Note that for any $\alpha > 0$, as $n\rightarrow \infty,$

In fact,

Since $n\Delta_n^{1+\beta}\rightarrow \infty$ for some $\beta > 0$ as $n\rightarrow \infty$, we have

In addition,

Thus (2.19) is obtained from (2.20)–(2.22).

By (2.19), we have, for any $\delta > 0,$

Hence (2.18) becomes

By Lemma 2.2, there exists a random variable $\xi_\alpha$ such that

for all $n\in {\textbf N}$. Moreover, ${\textbf E}|\xi_\alpha|^p < \infty$ for all $p\geq 1$. Therefore (2.14) holds.

Finally, we can get (2.15) using (2.13), (2.14), (2.10) and $\lim_{n\rightarrow \infty}\frac{\Delta_n}{e^{2\theta\Delta_n}-1} = \frac{1}{2\theta}$. The proof of Lemma 2.3 is finished.

3.   Proofs of Theorems

In this section, we will give proofs of Theorems 1.1-1.3.

By (2.7), we can rewrite $\hat{\theta}_n$ and $\tilde{\theta}_n$ as follows, respectively,

where

and

Proof of Theorem 1.1. For (1.6), according to (2.15) and (3.1), it suffices to prove that, as $n\rightarrow \infty,$

By the Minkowski inequality and (2.16), we have

where $c$ is a generic constant, and the last inequality also comes from the fact: as $x\rightarrow 0$,

Using similar arguments as the proof of (2.14), we can obtain (3.3). Thus (1.6) holds.

We can easily obtain (1.7) from (3.2), (2.15) and $\lim_{t\rightarrow \infty}\eta_t = \eta_{\infty}$ a.s., and this completes the proof of Theorem 1.1.

Proof of Theorem 1.2. (1) Firstly, we consider the case $q\geq \frac{1}{2}$. By (3.1), we get

Since $e^{\theta\Delta_n}-1-\theta\Delta_n\sim \frac{\theta^2}{2}\Delta_n^2$ as $\Delta_n\rightarrow 0,$ we obtain

Because $n\Delta_n^{1+\beta}\rightarrow \infty$ for some $\beta > 0$ as $n\rightarrow \infty$ and

Then, we have

On the other hand, by (3.4), we get, as $n\rightarrow \infty,$

since $\gamma > \frac{1}{2}$. Consequently, by (3.5)–(3.7) and (2.15), for any $q\geq \frac{1}{2}$, $\Delta_n^qe^{\theta T_n}(\hat{\theta}_n-\theta)$ is not tight.

For the case $0\leq q < \frac{1}{2}$, note that

and $\lim_{n\rightarrow \infty}\Delta_n^{q-\frac{1}{2}} = \infty$ and the previous case $q = \frac{1}{2}$. The proof of (1.8) is finished.

(2) By (3.1), it is obvious that

Since $n\Delta_n^3\rightarrow 0$ and $\frac{e^{\theta\Delta_n}-1-\theta\Delta_n}{\Delta_n^2}\rightarrow \frac{\theta^2}{2}$, we have, as $n\rightarrow \infty,$

Furthermore, by (3.4), we get, as $n\rightarrow \infty,$

By (3.8)–(3.10) and (2.15), we deduce (1.9). Hence, we complete the proof of Theorem 1.2.

Proof of Theorem 1.3. (1) Fix $q\geq \frac{1}{2},$ by (3.2) and (2.13), we obtain

By (2.16), we have, as $n\rightarrow \infty,$

We also get

since $\lim_{n\rightarrow \infty}n\Delta_n^{1+\beta} = \infty$ for some $\beta > 0,$ $\lim_{n\rightarrow \infty}\frac{e^{\theta T_n}}{T_n^{\frac{q+1}{\beta}}} = \infty$, $\lim_{n\rightarrow \infty}\frac{e^{2\theta\Delta_n}-1-2\theta\Delta_n}{\Delta_n^2} = 2\theta^2$ and $\lim_{n\rightarrow \infty}\frac{\Delta_n}{e^{2\theta\Delta_n}-1} = \frac{1}{2\theta}.$

Moreover, by (2.18), we deduce, as $n\rightarrow \infty,$

where $c(2)$ denotes $c(p) = c(2)$ in (2.18). Combining (3.11)–(3.14) and (2.15), we conclude that for every $q\geq \frac{1}{2},$ $\Delta_n^qe^{\theta T_n}(\tilde{\theta}_n-\theta)$ is not tight.

For the case $0\leq q < \frac{1}{2},$ we obtain it similarly to the proof of (1) in Theorem 1.2. Hence (1.10) holds.

(2) By (3.2) and (2.13), note that $T_n = t_n = n\Delta_n$, we can write

Similarly to (3.12)–(3.14), we obtain, as $n\rightarrow \infty,$

and

By (3.15)–(3.18), we can easily get (1.11).

Thus, we finish the proof of Theorem 1.3.

4.   Applications to fractional Gaussian processes

This section is devoted to some examples of the Gaussian process $G$. For example fractional Brownian motion, sub-fractional Brownian motion, bifractional Brownian motion and sub-bifractional Brownian motion.

4.1.   Fractional Brownian motion

The fractional Brownian motion (fBm) $B^{H} = \left\{B^{H}_t, t\geq 0 \right\}$ with Hurst parameter $H\in (0, 1)$ is defined as a centered Gaussian process starting from zero with covariance

Note that, when $H = \frac{1}{2}$, $B^{\frac{1}{2}}$ is a standard Brownian motion. By (4.1), we get

It is well-known that the self-similar index of fBm is $H$. Thus when $\frac{1}{2} < H < 1$, fBm satisfies the assumption (1.5). Hence Theorems 1.1–1.3 hold for the fBm $B^H (\frac{1}{2} < H < 1)$, namely, corresponding to Theorems 4, 6, and 7 in EI Onsy et al. (2019).

4.2.   Sub-fractional Brownian motion

The sub-fractional Brownian motion (sfBm) $S^{H} = \left\{S^{H}_t, t\geq 0 \right\}$ with Hurst parameter $H\in (0, 1)$ is defined as a centered Gaussian process starting from zero with covariance

Note that, when $H = \frac{1}{2}$, $S^{\frac{1}{2}}$ is a standard Brownian motion. The self-similar index for sfBm is also $H$. For more on sub-fractional Brownian motion, we can see Kuang and Xie (2015, 2017), Kuang and Liu (2015, 2018) and so on.

By (4.3), we have

When $\frac{1}{2} < H < 1$, by (4.4), we get

which shows that the assumption (1.5) holds for sfBm. Hence Theorems 1.1–1.3 hold for the sfBm $S^H (\frac{1}{2} < H < 1)$.

4.3.   Bifractional Brownian motion

The bifractional Brownian motion (bfBm) $B^{H, K} = \left\{B^{H, K}_t, t\geq 0 \right\}$ with parameters $H\in (0, 1)$ and $K\in (0, 1]$ is defined as a centered Gaussian process starting from zero with covariance

Note that, the case $K = 1$ corresponds to the fBm with Hurst parameter $H$. The self-similar index of bfBm is $HK$. By (4.6), we obtain

Since $K\leq 1$, when $1 < 2HK < 2$, we have

The assumption (1.5) holds for the bfBm $B^{H, K}(1 < 2HK < 2)$. Thus we also obtain Theorems 1.1-1.3 for the bfBm $B^{H, K}(1 < 2HK < 2)$.

4.4.   Sub-bifractional Brownian motion

El-Nouty and Journ$\acute{e}$ (2013) introduced the process $S^{H, K} = \{S^{H, K}_t, t\geq 0\}$ with indices $H\in (0, 1)$ and $K\in (0, 1],$ named the sub-bifractional Brownian motion (sbfBm) and defined as follows:

where $\{B^{H, K}_t, t\in \textbf{R}\}$ is a bifractional Brownian motion (bfBm) with indices $H\in (0, 1)$ and $K\in (0, 1].$ Clearly, the sbfBm is a centered Gaussian process such that$S^{H, K}_0 = 0$, with probability 1, and $\mbox{Var}\left(S^{H, K}_t\right) = \left(2^{K}-2^{2HK-1}\right)t^{2HK}.$ Note that since $(2H-1)K-1 < K-1\leq 0,$ it follows that $2HK-1 < K$. We can easily verify that $S^{H, K}$ is self-similar with index $HK$. When $K = 1$, $S^{H, 1}$ is the sub-fractional Brownian motion (sfBm). Straightforward computations show that for all $s, t\geq 0,$

and

where

So $S^{H, K}$ has $(HK-\epsilon)$-Hölder continuous paths for any $\epsilon\in (0, HK)$ thanks to Kolmogorov's continuity criterion. Kuang (2019) studied the collision local time of sub-bifractional Brownian Motions. Kuang and Li (2020) obtained Berry-Esséen bounds and proved the almost sure central limit theorem for the quadratic variation of the sub-bifractional Brownian motion.

By (4.9), we have

Since $K\leq 1$, when $1 < 2HK < 2$, we have

which means the assumption (1.5) also holds for the sbfBm $S^{H, K}(1 < 2HK < 2)$. Hence Theorems 1.1–1.3 also hold for the sbfBm $S^{H, K}(1 < 2HK < 2)$.

5.   Numerical simulations

In this section, we firstly simulate the sample paths of the process $X$ given by (1.1). By (2.7) and (2.8), we have

Let $t_i = i\Delta_n, i = 0, 1, \cdots, n, X_{t_0} = 0$, then,

for $i = 1, 2, \cdots, n.$

Let

where $\xi_i(i = 0, 1, \cdots, n)$ are standard normal random variables.

We know that

and

Thus, we can obtain the simulations of $\hat{\theta}_n$ and $\tilde{\theta}_n$.

Case 1: if $G$ is fBm $B^{H}(\frac{1}{2} < H < 1)$, then (5.1) and (5.2) become (5.3) and (5.4), respectively,

and

Case 2: if $G$ is sfBm $S^{H}(\frac{1}{2} < H < 1)$, then (5.1) and (5.2) become (5.5) and (5.6), respectively,

and

Case 3: if $G$ is bfBm $B^{H, K}(1 < 2HK < 2)$, then (5.1) and (5.2) are replaced by (5.7) and (5.8), respectively,

and

Case 4: if $G$ is sbfBm $S^{H, K}(1 < 2HK < 2)$, then (5.1) and (5.2) become (5.9) and (5.10), respectively,

and

Now we take $\Delta_n = 0.0002, n = 2\times 10^5$, and simulate 500 sample paths of $X$ for different values of $H, \theta$ or $H, K, \theta.$ The results of simulations are summarized in Tables 1–4 respectively. From these results, it can be seen that the mean of both constructed parameter estimators are close to the true parameter values and the standard deviations of $\hat{\theta}_n$ and $\tilde{\theta}_n$ are very small. Hence the numerical simulations confirm the theoretical research.

6.   Conclusions

EI Onsy et al. (2018) considered the parameter estimation for discretely observed nonergodic fractional Ornstein-Uhlenbeck process of the second kind. We extend their results to the case of Gaussian Ornstein-Uhlenbeck process of the second kind. We prove that two least squares type estimators are strongly consistent and rate consistent. Moreover, we give the numerical simulations which confirm the theoretical results. In the future, we will extend our results to more general Ornstein-Uhlenbeck type models such as Barndorff-Nielsen and Shephard type Ornstein-Uhlenbeck processes. On the other hand, we will consider the case of $0 < \gamma < \frac{1}{2}$ for the self-similar index $\gamma$ of Gaussian process $G$.

Acknowledgments

Nenghui Kuang was supported by the Natural Science Foundation of Hunan Province under Grant 2021JJ30233. Huantian Xie was supported in part by the NSF of Shandong Province (No.ZR2018LA008, 2019KJI003). The authors wish to thank both anonymous referees for careful reading of the previous versions of this paper and also their comments which improved the paper.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.