Fractional Langevin equation with $\alpha $-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.
