The class of distributions of periodic Ornstein-Uhlenbeck processes driven by Lévy processes
type: Item
instance of: scholarly article
title: The class of distributions of periodic Ornstein-Uhlenbeck processes driven by Lévy processes
zbMATH Open document ID: 1067.60014
author: Niranjan Chakravarthy
published in: Journal of Theoretical Probability
publication date: 2005-05-20T00:00:00Z
review text: Let \(Z\) be a Lévy process on \([0,1]\) with values in \(\mathbb R^d\) and \(c\in\mathbb R\setminus\{0\}\). Then the unique solution \(X\) to the Langevin equation \(dX_{t}=-cX_{t}dt + dZ_{t} \) for \(t\in[0,1]\) and \(X_{0}=X_{1}\) a.s. is called periodic Ornstein-Uhlenbeck process. Its canonic pathwise periodic extension on the real axis is a stationary process and in particular its stationary distribution satisfies \({\mathcal L}(X_{t})={\mathcal L}(X_{1})={\mathcal L}((e^c-1)^{-1}\int_{0}^{1}e^{cs}dZ_{s})\) on \([0,1]\). The authors are interested in the analysis of the class \(I(c)\) of stationary distributions of periodic Ornstein-Uhlenbeck processes with parameter \(c\) where the driving Lévy processes are ranging over all Lévy processes on \(d\)-dimensional Euclidean space. A usual Ornstein-Uhlenbeck process \(Y\) with parameter \(c>0\) on \([0,\infty)\) satisfies \({\mathcal L}(Y_{t})={\mathcal L}(Y_{0})\) for \(t\in[0,\infty)\) if and only if \({\mathcal L}(Z_{1})\) has finite \(\log\)-moment and \({\mathcal L}(Y_{0})={\mathcal L}(\int_{0}^{1}e^{-cs}dZ_{s})\). In this case the stationary distribution \({\mathcal L}(Y_{0})\) is selfdecomposable. Conversely, a selfdecomposable distribution \(\mu\) can be expressed in the same way with a Lévy process having finite \(\log\)-moment. The correspondence of \(\mu\) to \({\mathcal L}(Z_{1})\) is one to one for fixed \(c>0\). Thus the class of stationary distributions of usual Ornstein-Uhlenbeck processes does not depend on \(c\) and coincides with the class SD\((\mathbb{R}^d)\) of selfdecomposable distributions on \(d\)-dimensional Euclidian space.. For \(I(c)\) on the contrary the authors show that this class delicately depends on the parameter \(c\). They characterize \(I(c)\) in terms of properties of the Lévy measures of the distributions in this class. The mapping \(\varphi_{c}\) of \(\mu_{0}\) to \(\mu=\varphi_{c}(\mu_{0})={\mathcal L}(\int_{0}^{1}e^{cs}dZ_{s})\) is shown to be bijective and homeomorphic. Mo…
reviewed by: Andreas Martin
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G51
zbMATH DE Number: 2170060
zbMATH Keywords: stationary distributions
zbMATH Keywords: selfdecomposable distributions
MaRDI profile type: Publication
full work available at URL: s10959-004-2595-1
OpenAlex ID: W1969984666
cites work: Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. (With discussion)
cites work: Periodic Linear Differential Stochastic Processes
cites work: Periodic Ornstein-Uhlenbeck processes driven by Lévy processes
cites work: Spectral representations of infinitely divisible processes
cites work: Class L of multivariate distributions and its subclasses
cites work: Q4057878
DOI: 10.1007/S10959-004-2595-1
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description: scientific article; zbMATH DE number 2170060