Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series

Abstract

We introduce a fractional Langevin equation with α-stable noise and show that its solution {Yκ(t), t ≥ 0} is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of Y κ(t) via the measure of its codependence r(θ1, θ2, t). We prove that Yκ(t) is not a long-memory process in the sense of r(θ1, θ2, t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of the Langevin equation. © Instytut Matematyczny PAN, 2007.