Section: New Results
Markov characterization of the set-indexed Lévy process
Participant : Erick Herbin.
In collaboration with Prof. Ely Merzbach (Bar Ilan university, Israel).
In  , the class of set-indexed Lévy processes is considered using the stationarity property defined for the set-indexed fractional Brownian motion in  . The general framework of Ivanoff-Merzbach allows to consider standard properties of stochastic processes (e.g. martingale and Markov properties) in the set-indexed context. Processes are indexed by a collection of compact subsets of a metric space equipped with a Radon measure , which satisfies several stability conditions. Each process is assumed to admit an increment process defined as an additive extension of to the collections and
A set-indexed process is called a set-indexed Lévy process if the following conditions hold
On the contrary to previous works of Adler and Feigin (1984) on one hand, and Bass and Pyke (1984) one the other hand, the increment stationarity property allows to obtain explicit expressions for the finite-dimensional distributions of a set-indexed Lévy process. From these, we obtained a complete characterization in terms of Markov properties.
Among the various definitions for Markov property of a SI process, we considered the -Markov property. A collection of functions
where are s.t. , is called a transition system if the following conditions are satisfied:
A transition system is said
A set-indexed process is called -Markov if ,
where is the minimal filtration of the process .
Balan-Ivanoff (2002) proved that any SI process with independent increments is a -Markov process with a spatially homogeneous transition system. The following result proved in  shows that the converse is true.
Theorem Let be a set-indexed process with definite increments. The two following assertions are equivalent:
This result is strengthened in the following characterization of set-indexed Lévy processes as Markov processes with homogeneous transition systems.
Theorem Let be a set-indexed process with definite increments and satisfying the stochastic continuity property.
The two following assertions are equivalent:
Consequently, if is a transition system which is both spatially homogeneous and -homogeneous, then there exists a set-indexed process which is a -Markov process.