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

Tom 181 / 2007

M. Magdziarz, A. Weron Studia Mathematica 181 (2007), 47-60 MSC: 60G10, 60G52, 60H10, 62M10. DOI: 10.4064/sm181-1-4

Streszczenie

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.

Autorzy

  • M. MagdziarzHugo Steinhaus Center
    Institute of Mathematics and Computer Science
    Wroc/law University of Technology
    50-370 Wroc/law, Poland
    e-mail
  • A. WeronHugo Steinhaus Center
    Institute of Mathematics and Computer Science
    Wroc/law University of Technology
    50-370 Wroc/law, Poland
    e-mail

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