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

M. Magdziarz; A. Weron

M. Magdziarz ; A. Weron

Studia Mathematica, Tome 178 (2007), p. 47-60 / Harvested from The Polish Digital Mathematics Library

Résumé

We introduce a fractional Langevin equation with α-stable noise and show that its solution $Y_{κ} (t), t \geq 0$ 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 $Y_{κ} (t)$ 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.

Publié le : 2007-01-01

Zbl 1123.60045

EUDML-ID : urn:eudml:doc:285011

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     title = {Fractional Langevin equation with $\alpha$-stable noise. A link to fractional ARIMA time series},
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     year = {2007},
     pages = {47-60},
     zbl = {1123.60045},
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M. Magdziarz; A. Weron. Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series. Studia Mathematica, Tome 178 (2007) pp. 47-60. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm181-1-4/