Asymptotic Properties of Parameter Estimators in Fractional Vasicek Model

Abstract

We consider the fractional Vasicek model of the form dXt = (α-βXt)dt + γdBHt, driven by fractional Brownian motion BH with Hurst parameter H ∈ (0,1). We construct three estimators for an unknown parameter θ=(α,β) and prove their strong consistency.

Maximum Likelihood Estimation in the Fractional Vasicek Model

“…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.…”

“…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 dX t = κ(µ − X t ) dt + γ dB H t , and asymptotic theory for estimating only the persistent parameter κ was developed.…”

Maximum Likelihood Estimation in the Mixed Fractional Vasicek Model

“…The Vasicek model is used in finance, economics, biology, physics, chemistry, medicine , environmental studies and in other areas for modeling purposes. In a series of papers, Lohvinenko and Ralchenko [12,13,14,15] studied asymptotic properties of the maximum likelihood estimators of the parameters α and β in the fractional Vasicek model…”

“…Properties of fractional Vasicek model for modeling are investigated in Chronopoulu and Viens [2], Corlay et al [3], Hao et al [8], Song and Li [30] and Xiao et al [37] among others. Maximum likelihood estimation for fractional Vasicek model is investigated in Lohvinenko and Ralchenko [12,13,14,15] and Xiao and Yu [38,39].…”

Maximum likelihood estimation in the non-ergodic fractional Vasicek model

“…In the general case, the least squares and ergodic-type estimators of unknown parameters α and β were studied in [27,38,39]. The corresponding MLEs of α and β were presented in [25].…”