IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v28y2015i4d10.1007_s10959-014-0555-y.html
   My bibliography  Save this article

An Increment-Type Set-Indexed Markov Property

Author

Listed:
  • Paul Balança

    (École Centrale Paris)

Abstract

We present and study a Markov property, named $$\mathcal C$$ C -Markov, adapted to processes indexed by a general collection of sets. This new definition fulfils one important expectation for a set-indexed Markov property: there exists a natural generalization of the concept of transition operator which leads to characterization and construction theorems of $$\mathcal C$$ C -Markov processes. Several usual Markovian notions, including Feller and strong Markov properties, are also developed in this framework. Actually, the $$\mathcal C$$ C -Markov property turns out to be a natural extension of the two-parameter $$*$$ ∗ -Markov property to the multiparameter and the set-indexed settings. Moreover, extending a classic result of the real-parameter Markov theory, sample paths of multiparameter $$\mathcal C$$ C -Feller processes are proved to be almost surely right-continuous. Concepts and results presented in this study are illustrated with various examples.

Suggested Citation

  • Paul Balança, 2015. "An Increment-Type Set-Indexed Markov Property," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1271-1310, December.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0555-y
    DOI: 10.1007/s10959-014-0555-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-014-0555-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-014-0555-y?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. Erick Herbin & Ely Merzbach, 2009. "Stationarity and Self-Similarity Characterization of the Set-Indexed Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 22(4), pages 1010-1029, December.
    2. Zhou, Xiao-Wen & Zhou, Jian-Wei, 1993. "Simple function properties of two-parameter Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 47(1), pages 37-51, August.
    3. Nualart, D., 1983. "Two-parameter diffusion processes and martingales," Stochastic Processes and their Applications, Elsevier, vol. 15(1), pages 31-57, June.
    4. Herbin, Erick & Merzbach, Ely, 2013. "The set-indexed Lévy process: Stationarity, Markov and sample paths properties," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1638-1670.
    5. R. M. Balan & B. G. Ivanoff, 2002. "A Markov Property for Set-Indexed Processes," Journal of Theoretical Probability, Springer, vol. 15(3), pages 553-588, July.
    6. Balan, R. M., 2004. "Q-Markov random probability measures and their posterior distributions," Stochastic Processes and their Applications, Elsevier, vol. 109(2), pages 295-316, February.
    Full references (including those not matched with items on IDEAS)

    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. Hannebicque, Brice & Herbin, Érick, 2022. "Regularity of an abstract Wiener integral," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 154-196.
    2. Balan, R.M., 2007. "Markov jump random c.d.f.'s and their posterior distributions," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 359-374, March.
    3. R. Gutiérrez & C. Roldán & R. Gutiérrez-Sánchez & J. M. Angulo, 2007. "Prediction and Conditional Simulation of a 2D Lognormal Diffusion Random Field," Methodology and Computing in Applied Probability, Springer, vol. 9(3), pages 413-423, September.

    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:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0555-y. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.