Vasicek model is a type of 1-dimensional stochastic processes, it is used in various fields, such as economy, finance, environment. It was originally used to describe short-term interest rate fluctuations influenced by single market factors. Proposed by O. Vasicek , it is the first stochastic process model to describe the “mean reversion” characteristic of short-term interest rates. In the financial field, it can also be used as a random investment model in Wu et al. and Han et al..
Consider the Vasicek model driven by general Gaussian process, it satisfies the following Stochastic Differential Equation (SDE):
where and is a general one-dimensional centered Gaussian process that satisfies creftypecap 1.
This paper mainly focuses on the convergence rate of estimators of coefficient . Without loss of generality, we assume , then Vasicek model can be represent by the following form:
When the coefficients in the drift function is unknown, an important problem is to estimate the drift coefficients based on the observation. Based on the Brownian motion, Fergusson and Platen  present the maximum likelihood estimators of coefficients in Vasicek model. When the Vasicek model driven by the fractional Brownian motion, Xiao and Yu  consider the least squares estimators and their asymptotic behaviors. When , Hu and Nualart  study the moment estimation problem.
Since the Gaussian process mainly determines the trajectory properties of Vasicek model. Therefore, following the assumptions in Chen and Zhou , we make the following Hypothesis about .
Hypothesis 1 ( Hypothesis 1.1).
Let and , Covariance function of Gaussian process satisfies the following condition:
are constants independent with . Besides, for any .
The covariance functions of Gaussian processes such as fractional Brownian motion, subfractional Brownian motion and double fractional Brownian motion satisfy the above Hypothesis [7, Examples 1.5-1.8].
Proposition 1 ( (4) and (5)).
The estimator of is the continuous-time sample mean:
The second moment estimator of is given by
Following from Xiao and Yu , we present the LSE in Vasicek model.
Proposition 2 ( (7) and (8)).
The LSE is motivated by the argument of minimize a quadratic function of and :
Solving the equation, we can obtain the LSE of and , denoted by and respectively.
where the integral is an Itô-Skorohod integral.
Pei et al. prove the following consistencies and central limit theorems (CLT) of estimators.
Theorem 3 (, Theorem 1.2).
When creftypecap 1 is satisfied, both ME and LSE of are strongly consistent, that is
Theorem 4 (, Theorem 1.3).
Assume creftypecap 1 is satisfied. When is self-similar and , and are asymptotically normal as , that is,
Simalarly, is also asymptotically normal as :
We now present the main Theorems for the whole paper, and their details are given in the following sections.
Next, we show the convergence speed of mean coefficient estimators and .
Assume , and is a self-similar Gaussian process satisfying creftypecap 1 and . Then there exists a constant such that
In this section, we recall some basic facts about Malliavin calculus with respect to Gaussian process. The reader is referred to [15, 18, 17] for a more detailed explanation. Let be a continuous centered Gaussian process with and covariance function
defined on a complete probability space, where is generated by the Gaussian family . Denote as the the space of all real valued step functions on . The Hilbert space is defined as the closure of endowed with the inner product:
We denote as the isonormal Gaussian process on the probability space, indexed by the elements in , which satisfies the following isometry relationship:
The following Proposition shows the inner products representation of the Hilbert space .
Proposition 7 ( Proposition 2.1).
Denote as the set of bounded variation functions on . Then is dense in and
where is the Lebesgue-Stieljes signed measure associated with defined as
When the covariance function satisfies creftypecap 1,
Furthermore, the norm of the elements in can be induced naturally:
Remark ( Notation 1).
Let and be the constants given in creftypecap 1. For any , we define two norms as
For any in , define an operator from to to be
Proposition 8 ( Proposition 3.2).
Suppose that creftypecap 1 holds, then for any ,
and for any ,
Let and be the
-th tensor product and the-th symmetric tensor product of . For every , denote as the -th Wiener chaos of . It is defined as the closed linear subspace of generated by , where is the -th Hermite polynomial. Let such that , then for every and ,
where is the -th Wiener-Itô stochastic integral.
Denote as a complete orthonormal system in . The -th contraction between and is an element in :
The following proposition shows the product formula for the multiple integrals.
Proposition 9 ( Theorem 2.7.10).
Let and be two symmetric function. Then
where is the symmetrization of .
We then introduce the derivative operator and the divergence operator. For these details, see sections 2.3-2.5 of . Let be the class of smooth random variables of the form:
where , which partial derivatives have at most polynomial growth, and for , . Then, the Malliavin derivative of (with respect to ) is the element of defined by
Given and integer , let denote the closure of with respect to the norm
Denote (the divergence operator) as the adjoint of . The domain of is composed of those elements:
and is denoted by . If , then is the unique element of characterized by the duality formula:
We now introduce the infinitesimal generator of the Ornstein-Uhlenbeck semigroup. Let be a square integrable random variable. Denote as the orthogonal projection on the -th Wiener chaos . The operator is defined by . The domain of is
For any , define . is called the Pseudo-inverse of . Note that and holds for any .
The following Lemma 10 provides the Berry-Esséen upper bound on the sum of two random variables.
Lemma 10 ( Lemma 2).
For any variable and , the following inequality holds:
where is the standard Normal distribution function.
Using Malliavin calculus, Kim and Park  provide the Berry-Esséen upper bound of the quotient of two random variables.
Let be a zero-mean process, and satisfies a.s.. For simplicity, we define the following four functions:
Theorem 11 ( Theorem 2 and Corollary 1).
Let be a standard Normal variable. Assuming that for every , has an absolutely continuous law with respect to Lebesgue measure and , , as . Then, there exists a constant such that for large enough,
3. Berry-Esséen upper bounds of moment estimators
In this section, we will prove the Berry-Esséen upper bounds of Vasicek model moment estimators and . For the convenience of the following discussion, we first define :
where is a standard Normal variable. Next, we introduce the CLT of .
Theorem 12 ( Proposition 4.19).
Assume , and is a self-similar Gaussian process satisfying creftypecap 1 and . Then is asymptotically normal as :
is stochastic integral with respect to .
Following from the above Theorem, we can obtain the expanded form of (20):
Then, we can prove the convergence speed of .
Proof of formula (10).
Let , According to Lemma 10, we have
Since is self-similar, is standard Normal variable,
Following from Chebyshev inequality, we can obtain
The Proposition 3.10 of  ensures that is bounded. Combining the above results, we have
When is sufficiently large, there exists the constant such that the formula (10) holds. ∎
Similarly, we review the central limit theorem of .
Theorem 13 ( Proposition 4.18).
Assume and is a Gaussian process satisfying creftypecap 1. Then is asymptotically normal as :
The following Lemma shows the upper bound of the expectation of .
Let be the process defined by
When , there exists constant independent of such that
It is easy to see that
where is a constant. Also, we have
Combining the above two formulas, we obtain (23). ∎
Then we can obtain the Berry-Esséen upper bound of ME .
Proof of formula (8).
The following Lemma provides the upper bound of .
When is large enough, there exists constant such that
Since the Normal distribution is symmetric, we have
Consider the following processes:
where is an OU process driven by . According to  formula (63), we can obtain