Section: New Results
Sample paths properties 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  . Following Ivanoff-Merzbach's definitions of an indexing collection and its extensions and
a set-indexed process is called a set-indexed Lévy process if the following conditions hold
almost surely, where .
the increments of are independent: for all pairwise disjoint in , the random variables are independent.
has -stationary -increments, i.e. for all integer , all and for all increasing sequences and in , we have
is continuous in probability.
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 LÃ©vy process. From these, we obtained a complete characterization in terms of Markov properties.
The question of continuity is more complex in the set-indexed setting than for real-parameter stochastic processes. For instance, the set-indexed Brownian motion can be not continuous for some indexing collection. We consider a weaker form of continuity, which studies the possibility of point jumps.
The point mass jump of a set-indexed function at is defined by
and for each , denotes the collection of subsets with (a finite sub-semilattice which generates as ) and . A set-indexed function is said pointwise-continuous if , for all .
Theorem Let be a set-indexed Lévy process with Gaussian increments. Then for any such that , the sample paths of are almost surely pointwise-continuous inside , i.e.
In the general case, for all , For all with , we define
for all , the -field generated by the opened subsets of . The sample paths of the set-indexed Lévy processes can be derived from the following Lévy-Ito decomposition proved in  .
Theorem Let the generating triplet of the SI Lévy process .
Then can be decomposed as
is a set-indexed Lévy process with Gaussian increments, with generating triplet ,
is the set-indexed Lévy process with generating triplet , defined for some with by
and the processes and are independent.