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

Abstract

<abstract><p>Let $ B^{a, b}: = \{B_t^{a, b}, t\geq0\} $ be a weighted fractional Brownian motion of parameters $ a &gt; -1 $, $ |b| &lt; 1 $, $ |b| &lt; a+1 $. We consider a least square-type method to estimate the drift parameter $ \theta &gt; 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 &gt; -1 $, $ |b| &lt; 1 $, $ |b| &lt; a+1 $. Here we extend the results of ^{[<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>]} (resp. ^{[<xref ref-type="bibr" rid="b3">3</xref>]}), where the strong consistency and the asymptotic distribution of the estimators are proved for $ -\frac12 &lt; a &lt; 0 $, $ -a &lt; b &lt; a+1 $ (resp. $ -1 &lt; a &lt; 0 $, $ -a &lt; b &lt; a+1 $). Simulations are performed to illustrate the theoretical results.</p></abstract>