Fractional Langevin equation with $\alpha $-stable noise. A link to fractional ARIMA time series

M. Magdziarz¹ ; A. Weron¹

¹Hugo Steinhaus Center Institute of Mathematics and Computer Science Wroc/law University of Technology 50-370 Wroc/law, Poland

Studia Mathematica, Tome 181 (2007) no. 1, pp. 47-60

Résumé

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.

Détail

DOI : 10.4064/sm181-1-4

Keywords: introduce fractional langevin equation alpha stable noise its solution kappa geq stationary alpha stable ornstein uhlenbeck type process recently studied taqqu wolpert examine asymptotic dependence structure kappa via measure its codependence theta theta prove kappa long memory process sense theta theta however natural continuous time analogues fractional arima time series long memory framework langevin equation