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

Abstract

Let $ B^{a, b}: = \{B_t^{a, b}, t\geq0\} $ be a weighted fractional Brownian motion of parameters $ a > -1 $, $ |b| < 1 $, $ |b| < a+1 $. We consider a least square-type method to estimate the drift parameter $ \theta > 0 $ of the weighted fractional Ornstein-Uhlenbeck process $ X: = \{X_t, t\geq0\} $ defined by $ X_0 = 0; \ dX_t = \theta X_tdt+dB_t^{a, b} $. In this work, we provide least squares-type estimators for $ \theta $ based continuous-time and discrete-time observations of $ X $. The strong consistency and the asymptotic behavior in distribution of the estimators are studied for all $ (a, b) $ such that $ a > -1 $, $ |b| < 1 $, $ |b| < a+1 $. Here we extend the results of [1,2] (resp. [3]), where the strong consistency and the asymptotic distribution of the estimators are proved for $ -\frac12 < a < 0 $, $ -a < b < a+1 $ (resp. $ -1 < a < 0 $, $ -a < b < a+1 $). Simulations are performed to illustrate the theoretical results.

Keywords

• drift parameter estimation
• weighted fractional ornstein-uhlenbeck process
• strong consistency
• asymptotic distribution