Least squares estimation for non-ergodic weighted fractional Ornstein-Uhlenbeck process of general parameters

Let $B^{a,b}:=\{B_t^{a,b},t\geq0\}$ be a weighted fractional Brownian motion of parameters $a>-1$, $|b|<1$, $|b|<a+1$. We consider a least square-type method to estimate the drift parameter $\theta>0$ of the weighted fractional Ornstein-Uhlenbeck process $X:=\{X_t,t\geq0\}$ defined by $X_0=0; \ dX_t=\theta X_tdt+dB_t^{a,b}$. In this work, we provide least squares-type estimators for $\theta$ based continuous-time and discrete-time observations of $X$. The strong consistency and the asymptotic behavior in distribution of the estimators are studied for all $(a,b)$ such that $a>-1$, $|b|<1$, $|b|<a+1$. Here we extend the results of \cite{SYY2,SYY} (resp. \cite{CSC}), where the strong consistency and the asymptotic distribution of the estimators are proved for $-\frac12<a<0$, $-a<b<a+1$ (resp. $-1<a<0$, $-a<b<a+1$).


Introduction
Let B a,b := {B a,b t , t ≥ 0} be a weighted fractional Brownian motion (wfBm) with parameters (a, b) such that a > −1, |b| < 1 and |b| < a + 1, that is, B a,b is defined as a centered Gaussian process starting from zero with covariance The process B a,b was introduced by [2] as an extension of fractional Brownian motion (fBm) and sub-fractional Brownian motion (sfBm). Moreover, it shares several properties with fBm and sfBm, such as self-similarity, path continuity, behavior of increments, long-range dependence, non-semimartingale, and others. For more details we refer the reader to [2]. In this work we consider the non-ergodic Ornstein-Uhlenbeck process X := {X t , t ≥ 0} driven by a wfBm B a,b , that is the unique solution of the following linear stochastic differential equation where θ > 0 is an unknown parameter. An example of interesting problem related to (1.2) is the statistical estimation of θ when one observes X. In recent years, several researchers have been interested in studying statistical estimation problems for Gaussian Ornstein-Uhlenbeck processes. Let us mention some works in this direction in this case of Ornstein-Uhlenbeck process driven by a fractional Brownian motion B 0,b , that is, the solution of (1.2), where a = 0. In the ergodic case corresponding to θ < 0, the statistical estimation for the parameter θ has been studied by several papers, for instance [10,6,11,9,4] and the references therein. Further, in the non-ergodic case corresponding to θ > 0, the estimation of θ has been considered by using least squares method, for example in [5,1,7,8] and the references therein.
Here our aim is to estimate the the drift parameter θ based on continuous-time and discrete-time observations of X, by using least squares-type estimators (LSEs) for θ. First we will consider the following LSE as statistic to estimate θ based on the continuous-time observations {X s , s ∈ [0, t]} of (1.2), as t → ∞. We will prove the strong consistency and the asymptotic behavior in distribution of the estimator θ t for all parameters a > −1, |b| < 1 and |b| < a + 1. Our results extend those proved in [16,15], where − 1 2 < a < 0, −a < b < a + 1 only. Further, from a practical point of view, in parametric inference, it is more realistic and interesting to consider asymptotic estimation for (1.2) based on discrete observations. So, we will assume that the process X given in (1.2) is observed equidistantly in time with the step size ∆ n : t i = i∆ n , i = 0, . . . , n, and T n = n∆ n denotes the length of the "observation window". Then we will consider the following estimatorŝ as statistics to estimate θ based on the sampling data X t i , i = 0, . . . , n, as n −→ ∞. We will study the asymptotic behavior and the rate consistency of the estimatorsθ n andθ n for all parameters a > −1, |b| < 1 and |b| < a + 1. In this case, our results extend those proved in [3], where −1 < a < 0, −a < b < a + 1 only. The rest of the paper is organized as follows. In Section 2, we present auxiliary results that are used in the calculations of the paper. In Section 3, we prove the consistency and the asymptotic distribution of the estimator θ t given in (1.3), based on the continuous-time observations of X. In Section 3, we study the asymptotic behavior and the rate consistency of the estimatorsθ n andθ n defined in (1.4) and (1.5), respectively, based on the discretetime observations of X. We end the paper with a short review on some results from [5,7] needed for the proofs of our results.

Auxiliary Results
This section is devoted to prove some technical ingredients, which will be needed throughout this paper. In the following lemma we provide a useful expression for the covariance R a,b (t, s) of B a,b .
Lemma 2.1. Suppose that a > −1, |b| < 1 and |b| < a + 1. Then we can rewrite the covariance R a,b (t, s) of B a,b , given in (2.1) as where β(c, d) = Proof. We have for every s, t ≥ 0, Further, making change of variables x = u/t, we have for every t ≥ 0, Therefore, combining (2.3) and (2.4), we deduce that which proves (2.1).
We will also need the following technical lemma.
Lemma 2.2. We have as t −→ ∞, where Γ(.) is the standard gamma function, whereas the function m(t, s) is defined in (2.2).
Proof. We first prove (2.6). We have, On the other hand, by the change of variables where the latter equality comes from (2.6). Therefore the convergence (2.7) is proved.

LSE based on continuous-time observation
In this section we will establish the consistency and the asymptotic distribution of the least square-type estimator θ t given in (1.3), based on the continuous-time observation Recall that if X ∼ N (m 1 , σ 1 ) and Y ∼ N (m 2 , σ 2 ) are two independent random variables, then X/Y follows a Cauchy-type distribution. For a motivation and further references, we refer the reader to [14], as well as [12]. Notice also that if Theorem 3.1. Assume that a > −1, |b| < 1, |b| < a + 1, and let θ t be the estimator given in (1.3). Then, as t −→ ∞, Moreover, as t → ∞, is the standard Cauchy distribution with the probability density function 1 π(1+x 2 ) ; x ∈ R. Proof. In order to prove this Theorem 3.1, using Theorem 5.1, it suffices to check that the assumptions (H1), (H2), (H3), (H4) hold. It follows from (2.1) that for every 0 < s ≤ t, where the latter inequality comes from the change of variables x = (t − u)/(t − s). Hence, it is easy to see that there exists a constant C a,b depending only on a, b, such that Thus for all fixed T there exists a constant C a,b (T ) depending only on a, b, T such that for where we used the fact that a + b + 1 > 0. Therefore, using the fact that B a,b is Gaussian, and Kolmogorov's continuity criterion, we deduce that B a,b has a version with ((a + b + 1) ∧ (b + 1) − ε)-Hölder continuous paths for every ε ∈ (0, (a + b + 1) ∧ (b + 1)). Thus (H1) holds for any δ in (0, (a + b + 1) ∧ (b + 1)). On the other hand, according to (2.1) we have for every t ≥ 0, which proves that (H2) holds for γ = (a + b + 1)/2. Now it remains to check that the assumptions (H3) and (H4) hold for ν = −a/2 and σ B a,b = Γ(b+1) θ b+1 . Let us first compute the limiting variance of t −a/2 e −θt t 0 e θs dB a,b s as t → ∞.

LSEs based on discrete-time observations
In this section, our purpose is to study the asymptotic behavior and the rate consistency of the estimatorsθ n andθ n based on the sampling data X t i , i = 0, . . . , n of (1.2), where t i = i∆ n , i = 0, . . . , n, and T n = n∆ n denotes the length of the "observation window".  In addition, if we assume that n∆ 3 n → 0 as n → ∞, the estimatorsθ n andθ n are √ T n − consistent in the sense that the sequences T n (θ n − θ) and T n (θ n − θ) are tight.
Proof. In order to prove this Theorem 4.1, using Theorem 5.2, it suffices to check that the assumptions (H1), (H2), (H5) hold. From the proof of Theorem 3.1, the assumptions (H1), (H2) hold. Now it remains to check that (H5) holds. In this case, the process ζ is defined as whereas the integral is interpreted in the Young sense (see Appendix).
Using the formula (5.4) and (5.3), we can write Combining this with the fact for every if a > 0 together with ∆ n −→ 0, we deduce that there is a positive constant C such that if a > 0, which proves that the assumption (H5) holds. Therefore the desired result is obtained.

Appendix
Here we present some ingredients needed in the paper.
Let G = (G t , t ≥ 0) be a continuous centered Gaussian process defined on some probability space (Ω, F , P ) (Here, and throughout the text, we assume that F is the sigma-field generated by G). In this section we consider the non-ergodic case of Gaussian Ornstein-Uhlenbeck processes X = {X t , t ≥ 0} given by the following linear stochastic differential equation where θ > 0 is an unknown parameter. It is clear that the linear equation (5.1) has the following explicit solution whereas this latter integral is interpreted in the Young sense.
Let us introduce the following required assumptions.
(H3) There is constant ν in R such that the limiting variance of t ν e −θt t 0 e θs dG s exists as t → ∞, that is, there exists a constant σ G > 0 such that (H5) There exist positive constants ρ, C and a real constant µ such that . . , n, n ≥ 1.