Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index $H\in(0,\frac{1}{2})$

Abstract

We consider Langevin equation involving fractional Brownian motion with Hurst index H ∈ (0, 1 2 ). Its solution is the fractional Ornstein-Uhlenbeck process and with unknown drift parameter θ. We construct the estimator that is similar in form to maximum likelihood estimator for Langevin equation with standard Brownian motion. Observations are discrete in time. It is assumed that the interval between observations is n −1 , i.e. tends to zero (high frequency data) and the number of observations increases to infinity as n m with m > 1. It is proved that for positive θ the estimator is strongly consistent for any m > 1 and for negative θ it is consistent when m > 1 2H .60G22, 60F15, 60F25, 62F10, 62F12.

Hypothesis testing of the drift parameter sign for fractional Ornstein–Uhlenbeck process

“…Lemma 5.4 ( [12,13]). There exists a nonnegative random variable ζ such that for all s > 0, the following inequalities hold true: sup…”

“…It also implies that the inequality (22) holds for θ = 0, since in this case X t = x 0 +B H t . The bound (22) for θ < 0 was obtained in [13,Eq. (19)].…”

“…Note that in this case the estimator (5) coincides with the estimator (3), where the integral T 0 X t dX t is understood in the path-wise sense. In the papers [3,5,6,8,13,18] the discretized estimators were proposed.…”

Forecasting of time data with using fractional Brownian motion

“…The first motivation for studies of this process and its applications are considered in [2,17,18]. The results of studies of the properties of fractional Brownian motion and its application in models of natural and economic processes are covered in [3,4,[12][13][14][26][27][28]. Let's note the reviews [19,25].…”

A space-consistent version of the minimum-contrast estimator for linear stochastic evolution equations

“…Another interesting work on minimum-contrast estimation is [3], where a different version of the MCE based on fundamental-martingale technique for a real-valued fOU with Hurst index H > 1/2 is studied in the frameworks of both continuous-time and discrete-time sampling. Regarding drift parameter estimation for singular realvalued fOU processes (H < 1/2), recall the paper [12], where the LSE and the MCE for continuous-time observation are studied and the work [16], which investigates discrete-time version of the LSE.A modification of the minimum-contrast estimator (the weighted MCE) is introduced in this paper. Its construction benefits from the self-similarity property and it turns the time-consistent MCE (consistent with increasing time horizon) into a space-consistent estimator (fixed time horizon and increasing number of Fourier coordinates), which provides the best attainable speed of convergence in discrete-time setting.…”