The set-indexed Lévy process: Stationarity,Markov and sample paths properties |
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Authors: | Erick Herbin Ely Merzbach |
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Institution: | 1. Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France;2. Department of Mathematics, Bar Ilan University, 52900 Ramat-Gan, Israel |
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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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Keywords: | 60G10 60G15 60G17 60G51 60G57 60G60 60E07 60J65 |
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