Asymptotic Theory for Estimating Drift Parameters in the Fractional Vasicek Model

Abstract

This article develops an asymptotic theory for estimators of two parameters in the drift function in the fractional Vasicek model when a continuous record of observations is available. The fractional Vasicek model with long-range dependence is assumed to be driven by a fractional Brownian motion with the Hurst parameter greater than or equal to one half. It is shown that, when the Hurst parameter is known, the asymptotic theory for the persistence parameter depends critically on its sign, corresponding asymptotically to the stationary case, the explosive case, and the null recurrent case. In all three cases, the least squares method is considered, and strong consistency and the asymptotic distribution are obtained. When the persistence parameter is positive, the estimation method of Hu and Nualart (2010) is also considered.

Central limit theorems and minimum-contrast estimators for linear stochastic evolution equations

“…It was demonstrated there that asymptotic normality is violated for H > 3 4 . Similar approach used in more general setting of the fractional Vasicek model was presented in [26] and [27], where both ergodic and non-ergodic cases are considered. Recently, the 4 th moment theorem was successfully utilized to demonstrate not only asymptotic normality, but also to establish the speed of the convergence to the normal distribution (Berry-Esseentype of bounds) of the MC estimator of the drift parameter in one-dimensional SDEs driven by fBm -see [11] for discrete-time observations with increasing time-horizon and fixed mesh or [24] for continuous-time observations or discrete-time observations with combination of increasing time-horizon and observation frequency.…”

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].…”

Parameter Estimation for Fractional Diffusion Process with Discrete Observations

“…Prakasa Rao [4] gave an extensive review on most of the recent developments related to the parametric and other inference procedures for stochastic models driven by fBm. The latest study can be found in Xiao and Yu [5,6], who developed the asymptotic theory for least square estimators for two parameters in the drift function in the fractional Vasicek model with a continuous record of observations. Another possibility is to use Euler-type approximations for the solution of the above equation and to construct an MLE estimator based on the density of the observations given "the past", for the case of stochastic equations driven by Brownian motion.…”