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

Authors
Journal
Stochastic Processes and their Applications
0304-4149
Publisher
Elsevier
Volume
123
Issue
5
Identifiers
DOI: 10.1016/j.spa.2013.01.001
Keywords
  • Compound Poisson Process
  • Increment Stationarity
  • Infinitely Divisible Distribution
  • Lévy–Itô Decomposition
  • Lévy Processes
  • Markov Processes
  • Random Field
  • Independently Scattered Random Measures
  • Set-Indexed Processes
Disciplines
  • Mathematics

Abstract

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.

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