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The set-indexed Lévy process: Stationarity, Markov and sample paths properties

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  • Herbin, Erick
  • Merzbach, Ely

Abstract

We present a satisfactory definition of the important class of Lévy processes indexed by a general collection of sets. We use a new definition for increment stationarity of set-indexed processes to obtain different characterizations of this class. As an example, the set-indexed compound Poisson process is introduced. The set-indexed Lévy process is characterized by infinitely divisible laws and a Lévy–Khintchine representation. Moreover, the following concepts are discussed: projections on flows, Markov properties, and pointwise continuity. Finally the study of sample paths leads to a Lévy–Itô decomposition. As a corollary, the semi-martingale property is proved.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:5:p:1638-1670
    DOI: 10.1016/j.spa.2013.01.001
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    References listed on IDEAS

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    1. Dalang, Robert C. & Khoshnevisan, Davar, 2004. "Recurrent lines in two-parameter isotropic stable Lévy sheets," Stochastic Processes and their Applications, Elsevier, vol. 114(1), pages 81-107, November.
    2. (ed.), 1992. "Index," Books, Edward Elgar Publishing, number 1241.
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    4. Lagaize, Sandrine, 2001. "Hölder Exponent for a Two-Parameter Lévy Process," Journal of Multivariate Analysis, Elsevier, vol. 77(2), pages 270-285, May.
    5. Adler, R. J. & Monrad, D. & Scissors, R. H. & Wilson, R., 1983. "Representations, decompositions and sample function continuity of random fields with independent increments," Stochastic Processes and their Applications, Elsevier, vol. 15(1), pages 3-30, June.
    6. Balan, R. M., 2002. "Set-indexed processes with independent increments," Statistics & Probability Letters, Elsevier, vol. 59(4), pages 415-424, October.
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    Cited by:

    1. Paul Balança, 2015. "An Increment-Type Set-Indexed Markov Property," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1271-1310, December.
    2. Hannebicque, Brice & Herbin, Érick, 2022. "Regularity of an abstract Wiener integral," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 154-196.

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