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 ∈ R} is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied in [14]. We examine the asymptotic dependence structure of Yκ(t) via the measure of its codependence r(θ1; θ2; t) being the difference between the joint characteristic function of (Yκ(t), Yκ(0)) and the product of the characteristic functions of Yκ(t) and Yκ(0). We prove that Yκ(t) is not a long-memory process in the sense of codependence r(θ1; θ2; t). Moreover, we have found a natural continuous-time analogue of fractional ARIMA time series. 2000 Mathematics Subject Classification: 60G52.