Long-memory stable {O}rnstein-{U}hlenbeck processes
type: Item
instance of: scholarly article
title: Long-memory stable {O}rnstein-{U}hlenbeck processes
zbMATH Open document ID: 1087.60034
author: M. Maejima
author: Kenji Yamamoto
published in: Electronic Journal of Probability
publication date: 2005-03-08T00:00:00Z
full work available at URL: https://eudml.org/doc/126004
full work available at URL: paper19.abs.html
review text: The authors carry on work that has been done on stationary solutions of Langevin equations driven by noise processes that are more general than Brownian motion or fractional Brownian motion. If the driving process is Brownian motion \(W_t\), \(X_t = X_0 - \lambda \int_0^t X_s ds + W_t \), then, given \(X_0\), the solution is \[ X_t = e^{-\lambda t} [X_0 + \int_0^t e^{\lambda s} dW_s] = W_t + e^{-\lambda t} X_0 - \lambda \int_0^t e^{ \lambda s}W_s ds, \] and the initial condition \(X_0 = \int_{-\infty}^t e^{\lambda s} W_s ds \) yields the unique stationary solution \( X_t = W_t + \int_{-\infty}^t e^{- \lambda (t-s)} W_s ds \). The analogous result is shown to be true, if \(W_t\) is replaced by the linear fractional stable motion \( \Delta_{H, \alpha}(t) \) with \( 1 < \alpha < 2\), \(1/\alpha < H \). Here \( \Delta_{H, \alpha}(t) \) is a non-Gaussian self-similar process with possibly infinite variance and dependent increments. The authors investigate the dependence structure of the stationary solution and prove as main result long-range dependence.
reviewed by: Volker Wihstutz
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60G18
Mathematics Subject Classification ID: 60G52
Mathematics Subject Classification ID: 60G99
zbMATH DE Number: 2142115
zbMATH Keywords: dependent increments
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: independent increments
zbMATH Keywords: Levy process
zbMATH Keywords: Langevin equation
zbMATH Keywords: linear fractional stable motion
zbMATH Keywords: stationary process
zbMATH Keywords: stochastic differential equation with additive noise
MaRDI profile type: Publication
OpenAlex ID: W2096303140
DOI: 10.1214/EJP.V8-168
description: scientific article; zbMATH DE number 2142115