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

Jean Bertoin

Stochastic Processes and their Applications, 2019, vol. 129, issue 4, 1443-1454

Abstract: This work concerns the Ornstein–Uhlenbeck type process associated to a positive self-similar Markov process (X(t))t≥0 which drifts to ∞, namely U(t)≔e−tX(et−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 exponential functional Iˆ≔∫0∞exp(ξˆs)ds, where ξˆ is the dual of the real-valued Lévy process ξ related to X by the Lamperti transformation. This invariant measure is infinite (i.e. U is null-recurrent) if and only if ξ1∉L1(P). In that case, we determine the family of Lévy processes ξ for which U 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 ξ, and properties of time-substitutions based on additive functionals.

Keywords: Ornstein–Uhlenbeck type process; Stationarity; Self-similar Markov process; Lévy process; Exponential functional; Darling–Kac theorem (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:129:y:2019:i:4:p:1443-1454

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DOI: 10.1016/j.spa.2018.05.007

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