## 1. Introduction

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 [19], 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.[20] and Han et al.[9].

###### Definition 1.

Consider the Vasicek model driven by general Gaussian process, it satisfies the following Stochastic Differential Equation (SDE):

(1) |

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 [8] present the maximum likelihood estimators of coefficients in Vasicek model. When the Vasicek model driven by the fractional Brownian motion, Xiao and Yu [21] consider the least squares estimators and their asymptotic behaviors. When , Hu and Nualart [10] 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 [7], we make the following Hypothesis about .

###### Hypothesis 1 ([7] Hypothesis 1.1).

Let and , Covariance function of Gaussian process satisfies the following condition:

(2) |

where

are constants independent with . Besides, for any .

###### Remark.

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].

Assuming that there is only one trajectory , we can construct the least squares estimators (LSE) and the moment estimators (ME) (See [23, 22, 21, 3] for more details).

###### Proposition 1 ([6] (4) and (5)).

The estimator of is the continuous-time sample mean:

(3) |

The second moment estimator of is given by

(4) |

Following from Xiao and Yu [21], we present the LSE in Vasicek model.

###### Proposition 2 ([6] (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.

(5) |

(6) |

where the integral is an Itô-Skorohod integral.

Pei et al.[6] prove the following consistencies and central limit theorems (CLT) of estimators.

###### Theorem 3 ([6], Theorem 1.2).

When creftypecap 1 is satisfied, both ME and LSE of are strongly consistent, that is

###### Theorem 4 ([6], Theorem 1.3).

Assume creftypecap 1 is satisfied. When is self-similar and , and are asymptotically normal as , that is,

When ,

where

(7) |

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.

###### Theorem 5.

Let

be a standard Normal random variable, and

be the constant defined by (7). Assume and creftypecap 1 is satisfied. When is large enough, there exists a constant such that(8) | ||||

(9) |

where .

Next, we show the convergence speed of mean coefficient estimators and .

###### Theorem 6.

Assume , and is a self-similar Gaussian process satisfying creftypecap 1 and . Then there exists a constant such that

(10) | ||||

(11) |

## 2. Preliminary

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

(12) |

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:(13) |

We denote as the isonormal Gaussian process on the probability space, indexed by the elements in , which satisfies the following isometry relationship:

(14) |

The following Proposition shows the inner products representation of the Hilbert space [11].

###### Proposition 7 ([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,

(15) |

Furthermore, the norm of the elements in can be induced naturally:

###### Remark ([7] 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

(16) |

###### Proposition 8 ([7] Proposition 3.2).

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 ([15] Theorem 2.7.10).

Let and be two symmetric function. Then

(18) |

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 [15]. 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 ([4] Lemma 2).

For any variable and , the following inequality holds:

(19) |

where is the standard Normal distribution function.

Using Malliavin calculus, Kim and Park [12] 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 ([12] 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 :

(20) |

where is a standard Normal variable. Next, we introduce the CLT of .

###### Theorem 12 ([6] Proposition 4.19).

Assume , and is a self-similar Gaussian process satisfying creftypecap 1 and . Then is asymptotically normal as :

(21) |

where

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

where

The Proposition 3.10 of [6] ensures that is bounded. Combining the above results, we have

(22) |

When is sufficiently large, there exists the constant such that the formula (10) holds. ∎

Similarly, we review the central limit theorem of .

###### Theorem 13 ([6] 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 .

###### Lemma 14.

Let be the process defined by

When , there exists constant independent of such that

(23) |

###### Proof.

It is easy to see that

(24) |

where is a constant. Also, we have

(25) |

Combining the above two formulas, we obtain (23). ∎

Denote as

Then we can obtain the Berry-Esséen upper bound of ME .

###### Proof of formula (8).

The following Lemma provides the upper bound of .

###### Lemma 15.

When is large enough, there exists constant such that

where .

###### Proof.

Since the Normal distribution is symmetric, we have

Consider the following processes:

(27) | ||||

where is an OU process driven by . According to [6] formula (63), we can obtain

where