Parameter estimation for nonergodic Ornstein-Uhlenbeck process driven by the weighted fractional Brownian motion

Abstract

In this paper, we consider the nonergodic Ornstein-Uhlenbeck processdriven by the weighted fractional Brownian motion B a,b t with parameter a and b. Our goal is to estimate the unknown parameter θ > 0 based on the discrete observations of the process. We construct two estimatorsθ n andθ n of θ and show their strong consistency and the rate consistency.

Least squares estimation for non-ergodic weighted fractional Ornstein-Uhlenbeck process of general parameters

“…We will study the asymptotic behavior and the rate consistency of the estimators θn and θn for all parameters a > −1, |b| < 1 and |b| < a + 1. In this case, our results extend those proved in [3], where −1 < a < 0, −a < b < a + 1 only. The rest of the paper is organized as follows.…”

A Special Study of the Mixed Weighted Fractional Brownian Motion

“…To date, different algorithms have been used to simulate correlated Gaussian random variables (see, for instance, [21,22,30]). However, few works have studied the simulation of wfBm (see, for example, [8]). In our algorithm, we rely on the initial push outlined in Remark 2.3 of [6], in which the authors claimed that, in the case of b = 1, the process ξ can be represented through Bm B, as follows:…”

“…For a = 0, −1 < b < 1, wfBm corresponds to the celebrated fractional Brownian motion (fBm) with Hurst index b+1 2 , as well as to the well-known Brownian motion (Bm) when a = 0, b = 0. Many studies have been devoted to the weighted fractional Brownian motion and the related Ornstein-Uhlenbeck process, for instance [7][8][9].…”

“…In the ergodic case where θ < 0, several works, such as [11,12] and the references therein, have focused on the statistical estimation of the parameter θ. Furthermore, in the non-ergodic scenario corresponding to θ > 0, the estimation of parameter θ has been explored using the least squares method, as can be seen in [8,[13][14][15][16][17] and its references.…”

Maximum likelihood estimators of a long-memory process from discrete observations

“…Hu and Nualart [15] obtained least squares estimation (hereafter LSE) for the fractional OrnsteinUhlenbeck process and proposed the asymptotic normality of LSE using Malliavin calculus. More recently, there has been increased interest in studying asymptotic properties of LSE for the drift parameter in the univariate case with fractional processes (see, Azmoodeh and Morlanes [16]; Cheng et al [17]). Moreover, Xiao and Yu [18] and Xiao and Yu [19] considered the LSE in fractional Vasicek models in the stationary case, the explosive case, and the null recurrent case.…”