2011/08/03 by Érick Herbin, Herbin, Erick, Ely Merzbach +1
Economics, Econometrics and Finance · Mathematics · #60G10 #60G15 #60G17 #60G18 #60G51 #60G60 #Complex Systems and Time Series Analysis #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications
paper · doi:10.48550/arxiv.1108.0873
openalex publication_date 2011/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 semimartingale property is proved.