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## 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 [24] , the class of set-indexed Lévy processes is considered using the stationarity property defined for the set-indexed fractional Brownian motion in [23] . Following Ivanoff-Merzbach's definitions of an indexing collection $𝒜$ and its extensions ${𝒞}_{0}=\left\{U\setminus V;\phantom{\rule{0.277778em}{0ex}}U,V\in 𝒜\right\}$ and

$𝒞=\left\{U\setminus \bigcup _{1\le i\le n}{V}_{i};\phantom{\rule{0.277778em}{0ex}}n\in 𝐍;U,{V}_{1},\cdots ,{V}_{n}\in 𝒜\right\},$

a set-indexed process $X=\left\{{X}_{U};\phantom{\rule{0.277778em}{0ex}}U\in 𝒜\right\}$ is called a set-indexed Lévy process if the following conditions hold

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

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

3. $X$ has $m$-stationary ${𝒞}_{0}$-increments, i.e. for all integer $n$, all $V\in 𝒜$ and for all increasing sequences ${\left({U}_{i}\right)}_{i}$ and ${\left({A}_{i}\right)}_{i}$ in $𝒜$, we have

$\left[\forall i,\phantom{\rule{0.277778em}{0ex}}m\left({U}_{i}\setminus V\right)=m\left({A}_{i}\right)\right]⇒\left(\Delta {X}_{{U}_{1}\setminus V},\cdots ,\Delta {X}_{{U}_{n}\setminus V}\right)\stackrel{\left(d\right)}{=}\left(\Delta {X}_{{A}_{1}},\cdots ,\Delta {X}_{{A}_{n}}\right)$
4. $X$ 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 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 $x:𝒜\to 𝐑$ at $t\in 𝒯$ is defined by

 ${J}_{t}\left(x\right)=\underset{n\to \infty }{lim}\Delta {x}_{{C}_{n}\left(t\right)},\phantom{\rule{1.em}{0ex}}\mathrm{where}\phantom{\rule{1.em}{0ex}}{C}_{n}\left(t\right)=\bigcap _{\genfrac{}{}{0.0pt}{}{C\in {𝒞}_{n}}{t\in C}}C$ (12)

and for each $n\ge 1$, ${𝒞}_{n}$ denotes the collection of subsets $U\setminus V$ with $U\in {𝒜}_{n}$ (a finite sub-semilattice which generates $𝒜$ as $n\to \infty$) and $V\in {𝒜}_{n}\left(u\right)$. A set-indexed function $x:𝒜\to 𝐑$ is said pointwise-continuous if ${J}_{t}\left(x\right)=0$, for all $t\in 𝒯$.

Theorem Let $\left\{{X}_{U};\phantom{\rule{0.277778em}{0ex}}U\in 𝒜\right\}$ be a set-indexed Lévy process with Gaussian increments. Then for any ${U}_{max}\in 𝒜$ such that $m\left({U}_{max}\right)<+\infty$, the sample paths of $X$ are almost surely pointwise-continuous inside ${U}_{max}$, i.e.

$P\left(\forall t\in {U}_{max},{J}_{t}\left(X\right)=0\right)=1.$

In the general case, for all $ϵ>0$, For all $U\in 𝒜$ 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 $B\in {ℬ}_{ϵ}$, the $\sigma$-field generated by the opened subsets of $\left\{x\in 𝐑:|x|>ϵ\right\}$. The sample paths of the set-indexed Lévy processes can be derived from the following Lévy-Ito decomposition proved in [24] .

Theorem Let $\left(\sigma ,\gamma ,\nu \right)$ the generating triplet of the SI Lévy process $X$.

Then $X$ can be decomposed as

$\begin{array}{c}\hfill \forall \omega \in \Omega ,\forall U\in 𝒜,\phantom{\rule{1.em}{0ex}}{X}_{U}\left(\omega \right)={X}_{U}^{\left(0\right)}\left(\omega \right)+{X}_{U}^{\left(1\right)}\left(\omega \right),\end{array}$

where

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

2. ${X}^{\left(1\right)}=\left\{{X}_{U}^{\left(1\right)};\phantom{\rule{0.277778em}{0ex}}U\in 𝒜\right\}$ is the set-indexed Lévy process with generating triplet $\left(0,0,\sigma \right)$, defined for some ${\Omega }_{1}\in ℱ$ with $P\left({\Omega }_{1}\right)=1$ by

 $\begin{array}{cc}\hfill \forall \omega \in {\Omega }_{1},\phantom{\rule{4pt}{0ex}}& \forall U\in 𝒜,\hfill \\ \hfill {X}_{U}^{\left(1\right)}\left(\omega \right)& ={\int }_{|x|>1}x\phantom{\rule{4pt}{0ex}}{N}_{U}\left(dx,\omega \right)+\underset{ϵ↓0}{lim}{\int }_{ϵ<|x|\le 1}x\left[{N}_{U}\left(dx,\omega \right)-m\left(U\right)\right]\nu \left(dx\right),\hfill \end{array}$ (14)

where ${N}_{U}$ is defined in (13 ) and the last term of (14 ) converges uniformly in $U\subset {U}_{max}$ (for any given ${U}_{max}\in 𝒜$) as $ϵ↓0$,

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