Section: New Results
Sample paths properties of the setindexed Lévy process
Participant : Erick Herbin.
In collaboration with Prof. Ely Merzbach (Bar Ilan University, Israel).
In [24] , the class of setindexed Lévy processes is considered using the stationarity property defined for the setindexed fractional Brownian motion in [23] . Following IvanoffMerzbach's definitions of an indexing collection $\mathcal{A}$ and its extensions ${\mathcal{C}}_{0}=\{U\setminus V;\phantom{\rule{0.277778em}{0ex}}U,V\in \mathcal{A}\}$ and
a setindexed process $X=\left\{{X}_{U};\phantom{\rule{0.277778em}{0ex}}U\in \mathcal{A}\right\}$ is called a setindexed Lévy process if the following conditions hold

${X}_{{\varnothing}^{\text{'}}}=0$ almost surely, where ${\varnothing}^{\text{'}}={\bigcap}_{U\in \mathcal{A}}U$.

the increments of $X$ are independent: for all pairwise disjoint ${C}_{1},\cdots ,{C}_{n}$ in $\mathcal{C}$, the random variables $\Delta {X}_{{C}_{1}},\cdots ,\Delta {X}_{{C}_{n}}$ are independent.

$X$ has $m$stationary ${\mathcal{C}}_{0}$increments, i.e. for all integer $n$, all $V\in \mathcal{A}$ and for all increasing sequences ${\left({U}_{i}\right)}_{i}$ and ${\left({A}_{i}\right)}_{i}$ in $\mathcal{A}$, we have
$\left[\forall i,\phantom{\rule{0.277778em}{0ex}}m({U}_{i}\setminus V)=m\left({A}_{i}\right)\right]\Rightarrow (\Delta {X}_{{U}_{1}\setminus V},\cdots ,\Delta {X}_{{U}_{n}\setminus V})\stackrel{\left(d\right)}{=}(\Delta {X}_{{A}_{1}},\cdots ,\Delta {X}_{{A}_{n}})$ 
$X$ 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 finitedimensional distributions of a setindexed LÃ©vy process. From these, we obtained a complete characterization in terms of Markov properties.
The question of continuity is more complex in the setindexed setting than for realparameter stochastic processes. For instance, the setindexed 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 setindexed function $x:\mathcal{A}\to \mathbf{R}$ at $t\in \mathcal{T}$ is defined by
${J}_{t}\left(x\right)=\underset{n\to \infty}{lim}\Delta {x}_{{C}_{n}\left(t\right)},\phantom{\rule{1.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}{C}_{n}\left(t\right)=\bigcap _{\genfrac{}{}{0.0pt}{}{C\in {\mathcal{C}}_{n}}{t\in C}}C$  (12) 
and for each $n\ge 1$, ${\mathcal{C}}_{n}$ denotes the collection of subsets $U\setminus V$ with $U\in {\mathcal{A}}_{n}$ (a finite subsemilattice which generates $\mathcal{A}$ as $n\to \infty $) and $V\in {\mathcal{A}}_{n}\left(u\right)$. A setindexed function $x:\mathcal{A}\to \mathbf{R}$ is said pointwisecontinuous if ${J}_{t}\left(x\right)=0$, for all $t\in \mathcal{T}$.
Theorem Let $\{{X}_{U};\phantom{\rule{0.277778em}{0ex}}U\in \mathcal{A}\}$ be a setindexed Lévy process with Gaussian increments. Then for any ${U}_{max}\in \mathcal{A}$ such that $m\left({U}_{max}\right)<+\infty $, the sample paths of $X$ are almost surely pointwisecontinuous inside ${U}_{max}$, i.e.
In the general case, for all $\u03f5>0$, For all $U\in \mathcal{A}$ with $U\subset {U}_{max}$, we define
$\begin{array}{cc}\hfill {N}_{U}\left(B\right)& =\#\left\{t\in U:{J}_{t}\left(X\right)\in B\right\},\hfill \\ \hfill {X}_{U}^{B}& ={\int}_{B}x.{N}_{U}\left(dx\right),\hfill \end{array}$  (13) 
for all $B\in {\mathcal{B}}_{\u03f5}$, the $\sigma $field generated by the opened subsets of $\{x\in \mathbf{R}:x>\u03f5\}$. The sample paths of the setindexed Lévy processes can be derived from the following LévyIto decomposition proved in [24] .
Theorem Let $(\sigma ,\gamma ,\nu )$ the generating triplet of the SI Lévy process $X$.
Then $X$ can be decomposed as
where

${X}^{\left(0\right)}=\{{X}_{U}^{\left(0\right)};\phantom{\rule{0.277778em}{0ex}}U\in \mathcal{A}\}$ is a setindexed Lévy process with Gaussian increments, with generating triplet $(\sigma ,\gamma ,0)$,

${X}^{\left(1\right)}=\{{X}_{U}^{\left(1\right)};\phantom{\rule{0.277778em}{0ex}}U\in \mathcal{A}\}$ is the setindexed Lévy process with generating triplet $(0,0,\sigma )$, defined for some ${\Omega}_{1}\in \mathcal{F}$ with $P\left({\Omega}_{1}\right)=1$ by
$\begin{array}{cc}\hfill \forall \omega \in {\Omega}_{1},\phantom{\rule{4pt}{0ex}}& \forall U\in \mathcal{A},\hfill \\ \hfill {X}_{U}^{\left(1\right)}\left(\omega \right)& ={\int}_{\leftx\right>1}x\phantom{\rule{4pt}{0ex}}{N}_{U}(dx,\omega )+\underset{\u03f5\downarrow 0}{lim}{\int}_{\u03f5<\leftx\right\le 1}x\left[{N}_{U}(dx,\omega )m\left(U\right)\right]\nu \left(dx\right),\hfill \end{array}$ (14) where ${N}_{U}$ is defined in (13 ) and the last term of (14 ) converges uniformly in $U\subset {U}_{max}$ (for any given ${U}_{max}\in \mathcal{A}$) as $\u03f5\downarrow 0$,

and the processes ${X}^{\left(0\right)}$ and ${X}^{\left(1\right)}$ are independent.