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

Erick Herbin and Ely Merzbach

Stochastic Processes and their Applications, 2013, vol. 123, issue 5, 1638-1670

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.

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 (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:123:y:2013:i:5:p:1638-1670

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DOI: 10.1016/j.spa.2013.01.001

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