## 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.

Original language | English |
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Pages (from-to) | 1638-1670 |

Number of pages | 33 |

Journal | Stochastic Processes and their Applications |

Volume | 123 |

Issue number | 5 |

DOIs | |

State | Published - 2013 |

## Keywords

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