MAXIMUM LIKELIHOOD ESTIMATION IN THE FRACTIONAL VASICEK MODEL

where α,β,γ∈R+, and W is a standard Wiener process. From the financial point of view, β corresponds to the speed of recovery, the ratio α/β is the long-term average interest rate, and γ represents the stochastic volatility. Now the Vasicek model is widely used not only in finance, but also in various scientific areas such as economics, biology, physics, chemistry, medicine and environmental studies. The present paper deals with the fractional Vasicek model of the form


Introduction
The standard Vasicek model was proposed and studied by O. Vasicek [19] in 1977 for the purpose of interest rate modeling.It is described by the following stochastic differential equation where α, β, γ ∈ R + , and W is a standard Wiener process.From the financial point of view, β corresponds to the speed of recovery, the ratio α/β is the long-term average interest rate, and γ represents the stochastic volatility.Now the Vasicek model is widely used not only in finance, but also in various scientific areas such as economics, biology, physics, chemistry, medicine and environmental studies.The present paper deals with the fractional Vasicek model of the form where the Wiener process W is replaced with B H , a fractional Brownian motion with Hurst index H ∈ (1/2, 1).This generalization of the model (1.1) enables one to model processes with long-range dependence.Such processes appear in finance, hydrology, telecommunication, turbulence and image processing.In particular, various financial applications of the fractional Vasicek model (1.2) can be found in the articles [3][4][5][6][7][8][9]21].The goal of the paper is to construct maximum likelihood estimators (MLEs) for the unknown parameters α and β and to establish their consistency and asymptotic normality.We mention that the least squares and ergodic-type estimators in the fractional Vasicek model have been recently studied in [16] and [20].In [16] the strong consistency of these estimators was proved for the ergodic case β > 0, and the discretization of the ergodic-type estimators was considered.Note that in [20] a different parametrization was studied, namely and asymptotic theory for estimating only the persistent parameter κ was developed.The authors proved the strong consistency and asymptotic normality of the ergodic-type estimator for κ > 0. They also investigated the least squares estimator for the non-ergodic case κ < 0 and proved its convergence to the Cauchy distribution.This paper is organized as follows.In Section 2 we describe the model and give necessary definitions.In Section 3 we formulate and prove the main results on consistency and asymptotic normality of MLEs.Some auxiliary results are proved in the appendix.

Model description
Let (Ω, F, P) be a complete probability space.Let B H = {B H t , t ≥ 0} be a fractional Brownian motion on this probability space, that is, a centered Gaussian process with covariance function ) .
Throughout the paper we assume that H ∈ (1/2, 1).In what follows we consider the continuous (and even Hölder up to order H) modification of B H t that exists due to the Kolmogorov theorem.We study the fractional Vasicek model, described by the stochastic differential equation We assume that the parameters x 0 ∈ R, γ > 0 and H ∈ (1/2, 1) are known.The main goal is to estimate parameters α ∈ R and β > 0 by continuous observations of a trajectory of X on the interval [0, T ].We shall consider three problems: • estimation of α when β is known, • estimation of β when α is known, • estimation of unknown vector parameter θ = (α, β).
The equation (2.1) has a unique solution, which is given by where . Following [11], for 0 < s < t ≤ T we define Then the process M H is a Gaussian martingale, called the fundamental martingale, whose variance function ⟨ M H ⟩ is the function w H (see [17]).Moreover, the natural filtration of the martingale M H coincides with the natural filtration of the fractional Brownian motion B H . Define also three stochastic processes The process S is called a fundamental semimartingale [11].It has the following properties.
1.The process S is an (F t )-semimartingale with the decomposition 2. The process X admits the representation 3. Natural filtrations of processes S and X coincide.

Main results
Applying the analog of the Girsanov formula for a fractional Brownian motion ( [11, Theorem 3], see also [13]) and (2.3), one can obtain the following likelihood ratio (3.1) Now we can construct MLEs.

MLE for
It is unbiased, strongly consistent and normal: Proof.Let us maximize the likelihood ratio in (3.1) with respect to α.The first and the second partial derivatives are equal to Hence, the MLE for α is given by (3.2).By Lemma 2.1, the process S admits the representation: Hence Recall that the process M H is a martingale with quadratic variation w H . Since w H T → ∞, as T → ∞, by the strong law of large numbers for martingales [15, Theorem 2.6.10],we have Hence, α T a.s.
− − → α, as T → ∞, which confirms the strong consistency of the estimator.Since M H is a Gaussian process with variance function w H , it follows that Hence,

MLE for β when α is known
Theorem 3.2.Let H > 1/2 and α is known.The maximum likelihood estimator for β is It is strongly consistent and asymptotically normal: Proof.Let us maximize likelihood ratio in (3.1) with respect to β.We have Hence the MLE for β is given by (3.3).By Lemma 2.1, Hence Since the process M H is a martingale with quadratic variation w H , the process Therefore, by the strong law of large numbers for martingales [15, Theorem 2.6.10],we get the convergence β T a.s.− − → β, as T → ∞, which confirms the strong consistency of the estimator.Applying Lemma 4.7, we obtain Remark 3.3.If α = 0, then the process X is the fractional Ornstein-Uhlenbeck process.In this case the MLE . This MLE was first investigated in [12], where its strong consistency was established.Its asymptotic normality was proved in [18].

MLE for vector parameter (α, β)
Theorem 3.4.Let H > 1/2.The MLEs for α and β equal They are consistent and asymptotically normal: Proof.Let us maximize likelihood ratio in (3.1) with respect to α and β simultaneously.Obviously, the system of equations has the solution given by (3.6)-(3.7).Now we check the second partial derivatives of Λ H (T ): by the Cauchy-Schwarz inequality, which confirms maximization.
Applying the representation of the process S from Lemma 2.1 and formulas (3.4)-(3.5),we obtain Hence due to Lemmas 4.6 and 4.7 and the properties of the martingale M H we get which confirms asymptotical normality of the estimators and consequently their (weak) consistency.
Remark 3.6.The problem of finding the bivariate asymptotic distribution of the estimator ) is more involved and requires different tools.In particular, one should find the joint asymptotic distribution of the statistics S T , t , and This will be done in our further work.

Appendix
We start with the following simple lemma.
Lemma 4.1.For any H ∈ (0, 1) the following equation holds: Proof.The proof is carried out by substitution s = tz.
Let us introduce the following process Then U is a fractional Ornstein-Uhlenbeck process (see [2]), which is the solution of dU t = −βU t dt + dB H t , U 0 = 0. Maximum likelihood estimation for this process was widely studied in [12] and [18].Now from (2.2) one can get Then applying (4.1) and Lemma 4.1, we get where where I ν (z) is the modified Bessel function of the first kind.