Least Squares Estimator for Vasicek Model Driven by Fractional Lévy Processes

We construct least squares estimator for drift parameters based on time‐continuous observations, the consistency and asymptotic distribution of these estimators are studied in the non‐ergodic case. In contrast to the fractional Vasicek model, it can be regarded as a Lévy generalization of fractional Vasicek model.


1.Introduction
Statistical inference for stochastic differential equations is a main research direction in probability theory and its applications. When the noise is a standard Brownian motion, such problems have been extensively studied and some surveys and complete literatures for this direction could be found in Bishwal [2]. Moreover, since the seminal work of Vasicek [15], the Vasicek model driven by standard Brownian motion has been extensively applied in various fields, such as economics and finance, biology, physics, chemistry, medicine and environmental studies, where are unknown, the first term describes the so-called drift component The parameter determines the reversion speed of the stochastic component to their long-term mean . The economic interpretation of this mean-reversion component is that stochastic price fluctuations around the mean and price peaks are only temporarily, caused by for example power plant outages or capacity shortages. Indeed, when this model is used to describe some phenomena, it is important to identify the unknown parameters in this model. As a result, the parameter estimation problem for the Vasicek process driven by Brownian motion has played an important role in econometrics and becomes a interesting problem in the literature.
When the process degenerates into the well-known Ornstein-Uhlenbeck process. If the parameter is unknown and the process can be observed continuously, then an important problem is to estimate the parameter based on the (single path) observation . The most popular approaches are either the maximum likelihood estimators (MLE) or the least squares estimators (LSE), and in this case they coincide. For (ergodic case), the MLE of is asymptotically normal (Kutoyants [9]). For (non-ergodic case), the MLE of is asymptotically Cauchy (Dietz and Kutoyants [6]).
As an extension of Brownian motion, the fractional Brownian motion (fBm) has become an object of intensive study, due to its interesting properties and its applications in various scientific areas such as hydrology, telecommunications, fluid dynamics, turbulence, image processing, economics and finance. Recall that the fBm with Hurst index is the only centred Gaussian self-similar process with stationary increments, satisfies , , , the covariance function is given by It has stochastic integral representation in terms of a standard Brownian motion: where , is standard Brownian motion. For , coincides with the standard Brownian motion , but is neither a semi martingale nor a Markov process unless .
If the Brownian motion in the Vasicek model (1.1) is replaced with fBm, we get the following fractional Vasicek model (fVm) Parameter determines the persistence in . Depending on the sign of , the model can capture the stationary, the explosive, and the null recurrent behavior. The fVm was first used to describe the dynamics in volatility by Comte and Renault [3]. Other applications of fVm can be found in Comte, Coutin and Renault [4], Corlay, Lebovits and Véhel [5] references therein. Despite many applications of fVm in practice, estimation and the asymptotic theory in fVm has received little attention in the literature. Xiao and Yu [16] propose estimators for and develop the asymptotic theory for the estimators.

When
a very important special case of fVm is the fractional Ornstein-Uhlenbeck process. The parameter estimation for has been extensively studied using the MLE method (see Prakasa Rao [11]) or using the LSE technique (see Hu and Nualart [8]).
On the basis of sufficient study of fBm, many authors have proposed to use more general stochastic processes and random fields as stochastic models. Such applications have raised many interesting theoretical questions about stochastic processes and fields in general. Therefore, some generalizations of the fBm have been introduced such as sub-fractional Brownian motion, bifractional Brownian motion, weighted-fractional Brownian motion, fractional Lévy processes. However, in contrast to the extensive studies on fBm, there has been only a little systematic investigation on the statistical inference of other fractional processes. The main reason for this is the complexity of dependence structures. Recently, Fink and Klüppelberg [7] proved that the fractional Lévy driven Ornstein-Uhlenbeck processes (FLOUP) has unique stationary pathwise solution of the corresponding Langevin equation and the increments of an FLOUP exhibits long-range dependence.
However, there has been no study on parametric inference for Vasicek model with fractional Lévy noises yet. Motivated by the aforementioned works, as a first attempt, in this paper, we consider the generalized Vasicek model driven by fractional Lévy process (the precise definition is given below in Definition 2.1), and it is defined by the following stochastic differential equations In the present paper, we assume that the parameters and are unknown. We shall use the least square method to construct their estimators under the continuous observations, respectively. Our main results and aims are described as follows.
Firstly, we use the least square method to obtain the estimators of and . We introduce least squares estimators of and of the forms for all The two estimators are motivated by the following heuristic argument. By minimizing the contrast function (1.8) where denotes the differentiation of with respect to .
As a result, we can explicitly get the two least squares estimators and as follows for all , where the integral is interpreted as the Young integral (see, for example, Young [10]).
We shall prove the consistency of and , that is, and where the notation denotes "almost surely convergence".
The rest of this paper is organized as follows. In Section2, we present some preliminaries for lévy process and fractional lévy process. In Section3, we study the consistency of the least square's estimator and .

Lévy processes.
In this subsection, we mainly introduce the elementary properties of Lévy processes that will be used in following. More studies on the Lévy process can be found in Sato [13], Samorodnitsky and Taqqu [12] and the references therein.

Let be Lévy processes in without Brownian component. It is determined by its characteristic function in the Lévy-Khintchine form
where where and is the Lévy measure of on that satisfies This is a necessary and sufficient condition for to have finite mean and variance given by Furthermore, we restrict , then and Throughout this paper we will use a two side Lévy process constructed by taking two independent copies of a one-side Lévy process and setting (2.1)

Fractional Lévy processes.
In this subsection, we briefly recall the definition and properties of fractional Lévy process.
As an extension of fractional Brownian motion, fractional Lévy process is of interest in practical applications because of its stationarity of increments and long-range dependence. However, it is not Gaussian. Actually, the very large utilization of the fractional Brownian motion in practice (hydrology, telecommunications) are due to these properties (long range dependence). One prefers in general fractional Brownian motion before other processes because it is Gaussian and the calculus for it is easier. However, in concrete situations when the Gaussianity is not plausible for the model, one can use for example the fractional Lévy process. There exists a consistent literature that focuses on different theoretical and applications aspects of the fractional Lévy process. For example, Bender, Lindner and Schick' [1] studied the finite variation of fractional Lévy processes, Tikanmäki and Mishura [14] define fractional Lévy processes using the compact interval representation and proved that the fractional Lévy processes presented via different integral transformations have the same finite dimensional distributions if and only if they are fractional Brownian motions.
In this paper, we are interested in fractionally integrated processes. Therefore, we will work with the fractional integration parameter rather than the Hurst parameter . Moreover, we restrict ourselves to as we are interested in the long range dependence case. Based on the moving average integral representation of fractional Brownian motion, the class of fractional Lévy processes is introduced by replacing the Brownian motion by a general Lévy process with zero mean, finite variance and no Brownian component. where is symmetric -stable Lévy process (see Samorodnitsky and Taqqu [12]).
The following two Lemma gives an integral relationship between fractional lévy processes and integral with Lévy processes and the second-order property of the stochastic integral respect to fractional lévy processes.  .7) and (2.8)

Asymptotic behavior of the least squares estimator
In this section, let , we consider the strong consistency of and . Moreover, we also investigate the asymptotic of the estimator for the long term mean. We first consider the term From (2.6) and L'Hôspital's rule, we can get It is easy to see that the long term mean of is . It follows from (1.9) and (1.10), we have  By the Young integral, we have Using (2.5) and Young integral we have (3.17) and