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Conditioned real self-similar Markov processes

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  • Kyprianou, Andreas E.
  • Rivero, Víctor M.
  • Satitkanitkul, Weerapat

Abstract

In recent work, Chaumont et al. (2013) showed that is possible to condition a stable process with index α∈(1,2) to avoid the origin. Specifically, they describe a new Markov process which is the Doob h-transform of a stable process and which arises from a limiting procedure in which the stable process is conditioned to have avoided the origin at later and later times. A stable process is a particular example of a real self-similar Markov process (rssMp) and we develop the idea of such conditionings further to the class of rssMp. Under appropriate conditions, we show that the specific case of conditioning to avoid the origin corresponds to a classical Cramér–Esscher-type transform to the Markov Additive Process (MAP) that underlies the Lamperti–Kiu representation of a rssMp. In the same spirit, we show that the notion of conditioning a rssMp to continuously absorb at the origin also fits the same mathematical framework. In particular, we characterise the stable process conditioned to continuously absorb at the origin when α∈(0,1). Our results also complement related work for positive self-similar Markov processes in Chaumont and Rivero (2007).

Suggested Citation

  • Kyprianou, Andreas E. & Rivero, Víctor M. & Satitkanitkul, Weerapat, 2019. "Conditioned real self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 954-977.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:3:p:954-977
    DOI: 10.1016/j.spa.2018.04.001
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    References listed on IDEAS

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    1. Chaumont, L., 1996. "Conditionings and path decompositions for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 39-54, November.
    2. Chybiryakov, Oleksandr, 2006. "The Lamperti correspondence extended to Lévy processes and semi-stable Markov processes in locally compact groups," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 857-872, May.
    3. ARJAS, Elja & SPEED, T.P., 1973. "Symmetric Wiener-Hopf factorisations in Markov additive processes," LIDAM Reprints CORE 134, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Kyprianou, Andreas E. & Rivero, Victor & Şengül, Batı, 2017. "Conditioning subordinators embedded in Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1234-1254.
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    Cited by:

    1. Arista, Jonas & Rivero, Víctor, 2023. "Implicit renewal theory for exponential functionals of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 262-287.

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