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

Studia Mathematica (2007)

- Volume: 181, Issue: 1, page 47-60
- ISSN: 0039-3223

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topM. Magdziarz, and A. Weron. "Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series." Studia Mathematica 181.1 (2007): 47-60. <http://eudml.org/doc/285011>.

@article{M2007,

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

author = {M. Magdziarz, A. Weron},

journal = {Studia Mathematica},

keywords = {long memory; fractional Langevin equation; -stable processes; Ornstein-Uhlenbeck process; fractional ARIMA time series},

language = {eng},

number = {1},

pages = {47-60},

title = {Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series},

url = {http://eudml.org/doc/285011},

volume = {181},

year = {2007},

}

TY - JOUR

AU - M. Magdziarz

AU - A. Weron

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

JO - Studia Mathematica

PY - 2007

VL - 181

IS - 1

SP - 47

EP - 60

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

LA - eng

KW - long memory; fractional Langevin equation; -stable processes; Ornstein-Uhlenbeck process; fractional ARIMA time series

UR - http://eudml.org/doc/285011

ER -

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