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

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

2011-08-03

arxiv.org/abs/1108.0873v2

Mathematics

Probability

30 pages

Scientific paper

We present a satisfactory definition of the important class of L\'evy 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 L\'evy process is characterized by infinitely divisible laws and a L\'evy-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 L\'evy-It\^o decomposition. As a corollary, the semimartingale property is proved.

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