# Parametric Estimation in the Vasicek-Type Model Driven by Sub-Fractional Brownian Motion
## 1 Introduction  
Statistical inference for stochastic equations is a main research direction in probability theory and its applications. When the noise is a standard Brownian motion or a Levy process, such problems have been extensively studied. Some surveys and complete literature for this direction could be found in [PERSON] [1], [PERSON] [2], [PERSON] [3], [PERSON] [4], [PERSON] [5], and the references therein. However, in contrast to the extensive studies on semimartingale types, other statistical inferences associated with some Gaussian processes are very limited, and a common denominator in all these works is that it is assumed that the equation admits only an unknown parameter. Let us consider the parameter estimates of the Vasicek-type model driven by a Gaussian process \(G\):  
\[dX_{t}=(\mu+\theta X_{t})dt+dG_{t},\quad t\geq 0, \tag{1}\]  
where \(\mu\in\mathbb{R}\), \(\theta\in\mathbb{R}^{+}\) are two parameters. When \(\mu=0\) and \(G\) is a fractional Brownian motion with Hurst index \(H\in(0,1)\), the question has been studied by many authors. We mention the works of [PERSON] et al. [6], [PERSON] [7], [PERSON] and [PERSON] [8], [PERSON] and [PERSON] et al. [9, 10], [PERSON] and [PERSON] [11], [PERSON] [12], and the references therein for results on parameter estimation of stochastic equations driven by the fractional Brownian motion (fBm). When \(G\) is not a fractional Brownian motion, the research for this question is very limited. For \(\mu=0\) and \(G\) a sub-fractional Brownian motion, [PERSON] [13] considered the least squares estimation of \(\theta\) and studied the consistency and asymptotic behavior. For \(\mu=0\) and \(G\) a Gaussian process, [PERSON] et al. [14] showed the strong consistency and the asymptotic distribution of the least squares estimator \(\hat{\theta}\) of \(\theta\) based on the properties of \(G\), and as some examples, the authors also studied the three Vasicek-type models driven by fractional Brownian motion, sub-fractional Brownian motion, and bi-fractional Brownian motion, respectively. Motivated by these above results and for simplicity, in this paper, we consider the least squares estimation of Equation (1) when \(G\) is a sub-fractional Brownian motion \(S^{H}\) with Hurst index \(H\in(\frac{1}{2},1)\) and both \(\mu\) and \(\theta>0\) are unknown. That is, the parameter estimation of the so-called Vasicek-type model driven by sub-fractional Brownian motion:  
\[dX_{t}=(\mu+\theta X_{t})dt+dS^{H}_{t},\quad t\geq 0, \tag{2}\]  
where \(S^{H}\) is a sub-fractional Brownian motion and \(\mu\in\mathbb{R}\), \(\theta\in\mathbb{R}^{+}\) are two unknown parameters. On the other hand, there exists still a practical motivation for studying the parameter estimation, that is to provide optional tools to understand volatility modeling in finance. In fact, any mean-reverting model in continuous or discrete observations can be regarded as a model for stochastic volatility. We can consult the research monograph [15] for this modeling idea. Since stochastic volatility is not observed for many financial markets and the sub-fractional Brownian motion is a process without ergodicity, the discussions on the parameter estimation based on discrete observations are beyond the scope of this article. For the sake of simplicity, we focus on tackling the least squares estimation of Equation (2) based on the so-called continuous observations. The so-called sub-fractional Brownian motion (sub-fBm in short) \(S^{H}=\{S^{H}_{t},t\geq 0\}\) with index \(H\in(0,1)\) is introduced by [PERSON] et al. [16], which arises from occupation time fluctuations of branching particle systems with the Poisson initial condition. It is a mean zero Gaussian process with \(S^{H}_{0}=0\) and:  
\[R_{H}(t,s)\equiv E\left[S^{H}_{t}S^{H}_{s}\right]=s^{2H}+t^{2H}-\frac{1}{2} \left[(s+t)^{2H}+|t-s|^{2H}\right] \tag{3}\]  
for all \(s,t\geq 0\). For \(H=1/2\), \(S^{H}\) coincides with the standard Brownian motion \(B\). Sub-fBm \(S^{H}\) is neither a semimartingale nor a Markov process unless \(H=1/2\).