# Parametric Estimation in the Vasicek-Type Model Driven by Sub-Fractional Brownian Motion
## 1 Introduction
The sub-fBm has many properties analogous to those of fractional Brownian motion such as self-similarity, long/short-range dependence, and Holder paths. However, it has no stationary increments. Moreover, it admits the estimates:  
\[[(2-2^{2H-1})\wedge 1](t-s)^{2H}\leq E\left[\left(S^{H}_{t}-S^{H}_{s}\right)^{ 2}\right]\leq[(2-2^{2H-1})\lor 1](t-s)^{2H}. \tag{4}\]  
More works for sub-fractional Brownian motion can be found in [PERSON] Y et al. [17; 18], [PERSON] and [PERSON] [19], [PERSON] and [PERSON] [20], [PERSON] and [PERSON] [21; 22], [PERSON] [23; 24; 25; 26], [PERSON] et al. [27; 28], and the references therein. On the other hand, in contrast to the extensive studies on fractional Brownian motion, there has been little systematic investigation on other self-Gaussian processes. The main reason for this is the complexity of dependence structures, and in general, these Gaussian processes have no stationary increments and the representation based on Wiener integral with respect to a Brownian motion. Therefore, it seems interesting to study the asymptotic behavior associated with other self-Gaussian processes. Now, we consider Equation (2) with \(\frac{1}{2}<H<1\) and \(\theta>0\). Clearly, we have:  
\[X_{t}=\frac{\mu}{\theta}(e^{\theta t}-1)+e^{\theta t}\int_{0}^{t}e^{-\theta s }dS^{H}_{s}\]  
for all \(t\geq 0\), and the trajectory of \(X\) is \(\gamma\)-Holder continuous for all \(\gamma<H\) (see Section 3). As an immediate result, we see that the Young integral \(\int_{0}^{T}X_{t}dX_{t}\) is well defined for all \(\frac{1}{2}<H<1\). Let now the system Equation (2) be observed continuously, and let \(H\) be known. By using the least squares method due to [PERSON] and [PERSON] [10], the least squares estimators of \(\theta\) and \(\mu\) can be motivated by minimizing the contrast function:  
\[\rho(\mu,\theta)=\int_{0}^{T}\left|\dot{X}_{t}-(\mu+\theta X_{t})\right|^{2}dt.\]Minimizing the above contrast function \((\mu,\theta)\mapsto\rho(\mu,\theta)\), we introduce estimators of \(\theta\) and \(\mu\) as follows:  
\[\hat{\theta}_{T}=\frac{T\int_{0}^{T}X_{s}dX_{s}-X_{T}\int_{0}^{T}X_{s}ds}{T\int_ {0}^{T}X_{s}{}^{2}ds-(\int_{0}^{T}X_{s}ds)^{2}} \tag{5}\]  
and:  
\[\hat{\mu}_{T}=\frac{1}{T}\left(X_{T}-\hat{\theta}_{T}\int_{0}^{T}X_{s}ds\right) =\frac{X_{T}\int_{0}^{T}X_{s}{}^{2}ds-\frac{1}{2}(X_{T})^{2}\int_{0}^{T}X_{s}ds }{T\int_{0}^{T}X_{s}{}^{2}ds-(\int_{0}^{T}X_{s}ds)^{2}}, \tag{6}\]  
where the stochastic integral \(\int_{0}^{T}X_{t}dX_{t}\) is a Young integral for \(\frac{1}{2}<H<1\). Our main statement is as follows:  
* The least squares estimators \(\hat{\theta}_{T}\) and \(\hat{\mu}_{T}\) are strong consistent, and we have: \[e^{\theta T}(\hat{\theta}_{T}-\theta)\longrightarrow\frac{2\theta\lambda_{H}} {\lambda_{H}-\vartheta_{H}}\cdot\frac{\xi}{\eta+\frac{\mu}{g}(\lambda_{H}- \vartheta_{H})^{-1}},\] \[T\left(\hat{\mu}_{T}-\mu-\frac{1}{T}S_{T}^{H}\right) \longrightarrow 2\lambda_{H}\xi,\] and: \[dX_{t}=(\mu+\theta X_{t})dt+dS^{H}_{t},\quad t\geq 0, \tag{2}\]0 in distribution, as \(T\) tends to infinity, where \(\xi,\eta\sim N(0,1)\) are mutually independent, \(\zeta\sim N(0,2-2^{2H-1})\), \(\lambda_{H}=H\Gamma(2H)\), and: \[dX_{t}=(\mu+\theta X_{t})dt+dS^{H}_{t},\quad t\geq 0, \tag{2}\]1  
This paper is organized as follows. In Section 2, we present some preliminaries for sub-fBm. In Section 3, we prove the consistence of \(\hat{\mu}_{T}\) and \(\hat{\theta}_{T}\). In Section 4, we investigate the asymptotic distribution of estimators \(\hat{\mu}_{T}\) and \(\hat{\theta}_{T}\).