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 [21] , the class of set-indexed Lévy processes is considered using the stationarity property defined for the set-indexed fractional Brownian motion in [20] . 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 $𝒜$ of compact subsets of a metric space $𝒯$ equipped with a Radon measure $m$, which satisfies several stability conditions. Each process $\{X_U;U\in 𝒜\}$ is assumed to admit an increment process $\{\Delta X_C;C\in 𝒞\}$ defined as an additive extension of $X$ to the collections ${𝒞}_0=\{U\setminus V;U,V\in 𝒜\}$ and

A set-indexed process $X=\left\{X_U;U\in 𝒜\right\}$ 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

   $\left[\forall i,m(U_i\setminus V)=m\left(A_i\right)\right]\Rightarrow (\Delta X_{U_1\setminus V},\cdots ,\Delta X_{U_n\setminus V})\stackrel{\left(d\right)}{=}(\Delta X_{A_1},\cdots ,\Delta X_{A_n})$

4. $X$ is continuous in probability: if ${\left(U_n\right)}_{n\in 𝐍}$ is a sequence in $𝒜$ such that

   $\overline{{\bigcup }_n{\bigcap }_{k\ge n}{U}_k}=\bigcap _n\overline{{\bigcup }_{k\ge n}{U}_k}=A\in 𝒜$

   then

   $\underset{n\to \infty }{lim}P\left\{|X_{U_n}-X_A|>ϵ\right\}=0$

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 $𝒬$-Markov property. A collection $𝒬$ of functions

where $U,V\in 𝒜\left(u\right)$ are s.t. $U\subseteq V$, is called a transition system if the following conditions are satisfied:

1. $Q_{U,V}(•,B)$ is a random variable for all $B\in ℬ\left(𝐑\right)$.

2. $Q_{U,V}(x,•)$ is a probability measure for all $x\in 𝐑$.

3. For all $U\in 𝒜\left(u\right)$, $x\in 𝐑$ and $B\in ℬ\left(𝐑\right)$, $Q_{U,U}(x,B)={\delta }_x\left(B\right)$.

4. For all $U\subseteq V\subseteq W\in 𝒜\left(u\right)$,

   $\begin{array}{c}\hfill {\int }_{𝐑}{Q}_{U,V}(x,dy)Q_{V,W}(y,B)=Q_{U,W}(x,B).\end{array}$

A transition system $𝒬$ is said

• spatially homogeneous if for all $U\subset V$,

  $\forall x\in 𝐑,\forall B\in ℬ\left(𝐑\right),Q_{U,V}(x,B)=Q_{U,V}(0,B-x);$

• $m$-homogeneous if $Q_{U,V}$ only depends on $m(V\setminus U)$,

  i.e. $\forall U,V,U^{\text{'}},V^{\text{'}}\in 𝒜\left(u\right)$ such that $U\subset V$ and $U^{\text{'}}\subset V^{\text{'}}$,

  $m(V\setminus U)=m(V^{\text{'}}\setminus U^{\text{'}})\Rightarrow Q_{U,V}=Q_{U^{\text{'}},V^{\text{'}}}.$

A set-indexed process $X:=\{X_U;U\in 𝒜\}$ is called $𝒬$-Markov if $\forall U,V\in 𝒜\left(u\right)$, $U\subseteq V$

where ${\left({ℱ}_U\right)}_{U\in 𝒜\left(u\right)}$ is the minimal filtration of the process $X$.

Balan-Ivanoff (2002) proved that any SI process with independent increments is a $𝒬$-Markov process with a spatially homogeneous transition system. The following result proved in [21] shows that the converse is true.

Theorem Let $X=\{X_U;U\in 𝒜\}$ be a set-indexed process with definite increments. The two following assertions are equivalent:

1. $X$ is a $𝒬$-Markov process with a spatially homogeneous transition system $𝒬$ ;

2. $X$ has independent increments.

This result is strengthened in the following characterization of set-indexed Lévy processes as Markov processes with homogeneous transition systems.

Theorem Let $X=\{X_U;U\in 𝒜\}$ be a set-indexed process with definite increments and satisfying the stochastic continuity property.

The two following assertions are equivalent:

1. $X$ is a set-indexed Lévy process ;

2. $X$ is a $𝒬$-Markov process such that $X_{\varnothing }=0$ and the transition system $𝒬$ is spatially homogeneous and $m$-homogeneous.

Consequently, if $𝒬$ is a transition system which is both spatially homogeneous and $m$-homogeneous, then there exists a set-indexed process $X$ which is a $𝒬$-Markov process.