Lévy Processes

This entry contains an overview of Lévy processes and their multidimensional extensions, which are processes having stationary, independent increments that are continuous in probability thereby having versions with almost surely right continuous sample paths. Special cases are Poisson processes, Brownian motions and compound Poisson processes. Infinite divisibility, which is closely related to Lévy processes, is defined and discussed as well as the reflected version of these processes and associated martingales, both in the one- and multidimensional cases. Markov additive processes, which are extensions of Lévy process, as well as their reflected versions and associated martingales are introduced as well.