Fractional Langevin equation with -stable noise. A link to fractional ARIMA time series
Volume 181 / 2007
Studia Mathematica 181 (2007), 47-60
MSC: 60G10, 60G52, 60H10, 62M10.
DOI: 10.4064/sm181-1-4
Abstract
We introduce a fractional Langevin equation with \alpha-stable noise and show that its solution \{ Y_\kappa(t),\, t\geq 0 \} is the stationary \alpha-stable Ornstein–Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of Y_\kappa(t) via the measure of its codependence r(\theta_1,\theta_2,t). We prove that Y_\kappa(t) is not a long-memory process in the sense of r(\theta_1,\theta_2,t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of the Langevin equation.