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

Erick Herbin, Ely Merzbach

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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.

Original languageEnglish
Pages (from-to)1638-1670
Number of pages33
JournalStochastic Processes and their Applications
Issue number5
StatePublished - 2013


  • Compound Poisson process
  • Increment stationarity
  • Independently scattered random measures
  • Infinitely divisible distribution
  • Lévy processes
  • Lévy-Itô decomposition
  • Markov processes
  • Random field
  • Set-indexed processes


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