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An Increment-Type Set-Indexed Markov Property

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  • 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
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    References listed on IDEAS

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    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. Nualart, D., 1983. "Two-parameter diffusion processes and martingales," Stochastic Processes and their Applications, Elsevier, vol. 15(1), pages 31-57, June.
    3. 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.
    4. 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.
    5. 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.
    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.
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