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Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes


Bertoin, Jean (2019). Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes. Stochastic Processes and their Applications, 129(4):1443-1454.

Abstract

This work concerns the Ornstein–Uhlenbeck type process associated to a positive self-similar Markov process $( X ( t ) )_{t \geq 0}$ which drifts to $\infty$, namely $U ( t )$= $e^{-1} X (e^t - 1 )$ . We point out that $U$ is always a (topologically) recurrent ergodic Markov process. We identify its invariant measure in terms of the law of the exponentialfunctional $\hat{I} $ = $\int\limits_0^\infty exp ( \hat{\varepsilon}) ds $, where $\hat{\varepsilon}$ is the dual of the real-valued Lévy process $\varepsilon$ related to $X$ by the Lamperti transformation. This invariant measure is infinite (i.e. $U$ is null-recurrent) if and only if $\varepsilon_1 \notin L^1 (\mathbb{P} )$ . In that case, we determine the family of Lévy processes $\varepsilon$ for $U$ which fulfills the conclusions of the Darling–Kac theorem. Our approach relies crucially on a remarkable connection due to Patie (Patie, 2008) with another generalized Ornstein–Uhlenbeck process that can be associated to the Lévy process $\varepsilon$, and properties of time-substitutions based on additive functionals.

Abstract

This work concerns the Ornstein–Uhlenbeck type process associated to a positive self-similar Markov process $( X ( t ) )_{t \geq 0}$ which drifts to $\infty$, namely $U ( t )$= $e^{-1} X (e^t - 1 )$ . We point out that $U$ is always a (topologically) recurrent ergodic Markov process. We identify its invariant measure in terms of the law of the exponentialfunctional $\hat{I} $ = $\int\limits_0^\infty exp ( \hat{\varepsilon}) ds $, where $\hat{\varepsilon}$ is the dual of the real-valued Lévy process $\varepsilon$ related to $X$ by the Lamperti transformation. This invariant measure is infinite (i.e. $U$ is null-recurrent) if and only if $\varepsilon_1 \notin L^1 (\mathbb{P} )$ . In that case, we determine the family of Lévy processes $\varepsilon$ for $U$ which fulfills the conclusions of the Darling–Kac theorem. Our approach relies crucially on a remarkable connection due to Patie (Patie, 2008) with another generalized Ornstein–Uhlenbeck process that can be associated to the Lévy process $\varepsilon$, and properties of time-substitutions based on additive functionals.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Physical Sciences > Modeling and Simulation
Physical Sciences > Applied Mathematics
Uncontrolled Keywords:Modelling and Simulation, Statistics and Probability, Applied Mathematics
Language:English
Date:1 April 2019
Deposited On:23 Jan 2019 11:58
Last Modified:29 Jul 2020 07:58
Publisher:Elsevier
ISSN:0304-4149
OA Status:Closed
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1016/j.spa.2018.05.007

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