IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v129y2019i3p954-977.html
   My bibliography  Save this article

Conditioned real self-similar Markov processes

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414918300905
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2018.04.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ceren Vardar-Acar & Mine Çağlar & Florin Avram, 2021. "Maximum Drawdown and Drawdown Duration of Spectrally Negative Lévy Processes Decomposed at Extremes," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1486-1505, September.
    2. Chaumont, Loïc & Rivero, Víctor, 2007. "On some transformations between positive self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1889-1909, December.
    3. A. B. Dieker & M. Mandjes, 2011. "Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 221-267, June.
    4. Fernando Cordero, 2016. "The First Passage Time of a Stable Process Conditioned to Not Overshoot," Journal of Theoretical Probability, Springer, vol. 29(3), pages 776-796, September.
    5. Campi, Luciano & Cetin, Umut & Danilova, Albina, 2013. "Explicit construction of a dynamic Bessel bridge of dimension 3," LSE Research Online Documents on Economics 45263, London School of Economics and Political Science, LSE Library.
    6. Çağlar, M. & Kyprianou, A. & Vardar-Acar, C., 2022. "An optimal stopping problem for spectrally negative Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1109-1138.
    7. Lyu, Hanbaek & Sivakoff, David, 2019. "Persistence of sums of correlated increments and clustering in cellular automata," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1132-1152.
    8. Kyprianou, Andreas E. & Palau, Sandra & Saizmaa, Tsogzolmaa, 2021. "Attraction to and repulsion from a subset of the unit sphere for isotropic stable Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 272-293.
    9. Chaumont, L. & Kyprianou, A.E. & Pardo, J.C., 2009. "Some explicit identities associated with positive self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 980-1000, March.
    10. Kim, Panki & Song, Renming & Vondraček, Zoran, 2023. "Positive self-similar Markov processes obtained by resurrection," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 379-420.
    11. Döring, Leif & Kyprianou, Andreas E. & Weissmann, Philip, 2020. "Stable processes conditioned to avoid an interval," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 471-487.
    12. 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.
    13. Kyprianou, Andreas E. & Vakeroudis, Stavros M., 2018. "Stable windings at the origin," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4309-4325.
    14. Foss, Sergey G. & Puhalskii, Anatolii A., 2011. "On the limit law of a random walk conditioned to reach a high level," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 288-313, February.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:129:y:2019:i:3:p:954-977. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.