EN FR
EN FR

## Section: New Results

### An increment type set-indexed Markov property

Participant : Paul Balança.

[1] investigates a new approach for the definition of a set-indexed Markov property, named $𝒞$-Markov. The study is based on Merzbach and Ivanoff's set-indexed formalism, i.e. $𝒜$ denotes a set-indexed collection and $𝒞$ the family of increments $C=A\setminus B$, where $A\in 𝒜$ and $B\in 𝒜\left(u\right)$ (finite unions of sets from $𝒜$). Moreover, for any $C=A\setminus B$, $B={\cup }_{i=1}^{k}{A}_{i}$, ${𝒜}_{𝐂}$ is defined as the following subset of $𝒜$:

${𝒜}_{𝐂}=\left\{U\in {𝒜}_{\ell };\phantom{\rule{0.166667em}{0ex}}U⊈{B}^{\circ }\right\}:=\left\{{U}_{\phantom{\rule{-1.0pt}{0ex}}C}^{\phantom{\rule{0.2pt}{0ex}}1},\cdots ,{U}_{\phantom{\rule{-1.0pt}{0ex}}C}^{\phantom{\rule{0.2pt}{0ex}}p}\right\},\phantom{\rule{2.em}{0ex}}\text{where}\phantom{\rule{4.pt}{0ex}}p=|{𝒜}_{𝐂}|$

and ${𝒜}_{\ell }$ corresponds to the semilattice $\left\{{A}_{1}\cap \cdots \cap {A}_{k},\cdots ,{A}_{1}\cap {A}_{2},{A}_{1}\cdots ,{A}_{k}\right\}\subset 𝒜$. The notation ${𝐗}_{𝐂}$ refers to a random vector ${𝐗}_{𝐂}=\left({X}_{{U}_{\phantom{\rule{-1.0pt}{0ex}}C}^{\phantom{\rule{0.2pt}{0ex}}1}},\cdots ,{X}_{{U}_{\phantom{\rule{-1.0pt}{0ex}}C}^{\phantom{\rule{0.2pt}{0ex}}p}}\right)$. Similarly, ${𝐱}_{𝐂}$ is used to denote a vector of variables $\left({x}_{{U}_{\phantom{\rule{-1.0pt}{0ex}}C}^{\phantom{\rule{0.2pt}{0ex}}1}},\cdots ,{x}_{{U}_{\phantom{\rule{-1.0pt}{0ex}}C}^{\phantom{\rule{0.2pt}{0ex}}p}}\right)$.

Then, an $E$-valued set-indexed process ${\left({X}_{A}\right)}_{A\in 𝒜}$ is said to be $𝒞$-Markov with respect to a filtration ${\left({ℱ}_{A}\right)}_{A\in 𝒜}$ if it is adapted to ${\left({ℱ}_{A}\right)}_{A\in 𝒜}$ and if it satisfies

 $𝔼\left[\phantom{\rule{1.0pt}{0ex}}f\left({X}_{A}\right)\phantom{\rule{1.5pt}{0ex}}|\phantom{\rule{1.5pt}{0ex}}{𝒢}_{\phantom{\rule{-0.4pt}{0ex}}C}^{\phantom{\rule{0.5pt}{0ex}}*}\phantom{\rule{1.0pt}{0ex}}\right]=𝔼\left[\phantom{\rule{1.0pt}{0ex}}f\left({X}_{A}\right)\phantom{\rule{1.5pt}{0ex}}|\phantom{\rule{1.5pt}{0ex}}{𝐗}_{𝐂}\phantom{\rule{1.0pt}{0ex}}\right]\phantom{\rule{1.em}{0ex}}ℙ\text{-a.s.}$ (7)

for all $C=A\setminus B\in 𝒞$ and any bounded measurable function $f:E\to 𝐑$. The sigma-algebra ${𝒢}_{\phantom{\rule{-0.4pt}{0ex}}C}^{\phantom{\rule{0.5pt}{0ex}}*}$ is usually called the strong history of ${\left({ℱ}_{A}\right)}_{A\in 𝒜}$ and is defined as ${𝒢}_{\phantom{\rule{-0.4pt}{0ex}}C}^{\phantom{\rule{0.5pt}{0ex}}*}={\bigvee }_{A\in 𝒜,A\cap C=\varnothing }\phantom{\rule{0.166667em}{0ex}}{ℱ}_{A}$.

The $𝒞$-Markov approach has several advantages compared to existing set-indexed Markov literature (mainly $𝒬$-Markov described in [48] ). It appears to be a natural extension of the classic one-parameter Markov property. In particular, the concept of transition system can easily extended to our formalism: for any $𝒞$-Markov process $X$, one can defined $𝒫=\left\{{P}_{\phantom{\rule{-0.2pt}{0ex}}C}\left({𝐱}_{𝐂};\mathrm{d}{x}_{A}\right);C\in 𝒞\right\}$ as

$\forall {𝐱}_{𝐂}\in {E}^{|{𝒜}_{𝐂}|},\Gamma \in ℰ;\phantom{\rule{1.em}{0ex}}{P}_{\phantom{\rule{-0.2pt}{0ex}}C}\left({𝐱}_{𝐂};\Gamma \right):=ℙ\left({X}_{A}\in \Gamma \phantom{\rule{1.5pt}{0ex}}|\phantom{\rule{1.5pt}{0ex}}{𝐗}_{𝐂}={𝐱}_{𝐂}\right).$

A $𝒞$-transition system $𝒫$ happens to satisfy a set-indexed Chapman-Kolmogorov equation,

 $\forall C\in 𝒞,\phantom{\rule{0.166667em}{0ex}}{A}^{\text{'}}\in 𝒜;\phantom{\rule{2.em}{0ex}}{P}_{\phantom{\rule{-0.2pt}{0ex}}C}f={P}_{\phantom{\rule{-0.2pt}{0ex}}{C}^{\text{'}}}{P}_{\phantom{\rule{-0.2pt}{0ex}}{C}^{\text{'}\text{'}}}f\phantom{\rule{1.em}{0ex}}\text{where}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}{C}^{\text{'}}=C\cap {A}^{\text{'}}\text{,}\phantom{\rule{4.pt}{0ex}}{C}^{\text{'}\text{'}}=C\setminus {A}^{\text{'}}$ (8)

and $f$ is a bounded measurable function.

Similarly to the classic Markovian theory, is is proved in [1] that the initial distribution $\mu$ and $𝒫$ characterize entirely the law of a $𝒞$-Markov process, and that conversely, for any initial law and any $𝒞$-transition system, a corresponding canonical set-indexed $𝒞$-Markov process can be constructed. $𝒞$-Markov processes enjoy several other properties such as

1. Projections on elementary flows are Markovian;

2. Conditional independence of natural filtrations;

3. Strong Markov property.

The class of set-indexed Lévy processes defined and studied in [21] offers examples of $𝒞$-Markov processes whose transition probabilities correspond to

 $\forall C=A\setminus B\in 𝒞,\phantom{\rule{1.em}{0ex}}\forall \Gamma \in ℰ;\phantom{\rule{1.em}{0ex}}{P}_{\phantom{\rule{-0.2pt}{0ex}}C}\left({𝐱}_{𝐂};\Gamma \right)={\mu }^{m\left(C\right)}\left(\Gamma -\Delta {x}_{B}\right),$ (9)

where $m$ is a measure on $𝒯$ and $\mu$ the infinitely divisible probability measure that characterizes the Lévy process. We note that the transition system related the $𝒬$-Markov property has a different form, even if it is related.

Another non-trivial example of $𝒞$-Markov process is the set-indexed Ornstein-Uhlenbeck process that has been introduced and studied in [32] . It is a Gaussian Markovian process whose transition densities are given by

 ${p}_{\phantom{\rule{-0.2pt}{0ex}}C}\left({𝐱}_{𝐂};y\right)=\frac{1}{{\sigma }_{C}\sqrt{2\pi }}exp\left[-\frac{1}{2{\sigma }_{C}^{2}}{\left(y-{e}^{-\lambda m\left(A\right)}\left[\sum _{i=1}^{n}{\left(-1\right)}^{{\epsilon }_{i}}\phantom{\rule{0.166667em}{0ex}}{x}_{{U}_{\phantom{\rule{-1.0pt}{0ex}}C}^{\phantom{\rule{0.2pt}{0ex}}i}}\phantom{\rule{0.166667em}{0ex}}{e}^{\lambda m\left({U}_{\phantom{\rule{-1.0pt}{0ex}}C}^{\phantom{\rule{0.2pt}{0ex}}i}\right)}\right]\right)}^{2}\right],$ (10)

where $\lambda$ and $\sigma$ are positive parameters, $m$ is a measure on $𝒯$ and

${\sigma }_{C}^{2}=\frac{{\sigma }^{2}}{2\lambda }\left(1-{e}^{-2\lambda m\left(A\right)}\left[\sum _{i=1}^{n}{\left(-1\right)}^{{\epsilon }_{i}}{e}^{2\lambda m\left({U}_{\phantom{\rule{-1.0pt}{0ex}}C}^{\phantom{\rule{0.2pt}{0ex}}i}\right)}\right]\right).$

In the particular case of multiparameter processes, corresponding to the indexing collection $𝒜=\left\{\left[0,t\right];t\in {𝐑}_{+}^{N}\right\}$, the $𝒞$-Markov formalism is related to several existing works. It generalizes the two-parameter $*$-Markov property introduced in [53] and also embraces the multiparameter Markov property investigated recently in [68] . Finally, under some Feller assumption on the transition system, a multiparameter $𝒞$-Markov process is proved to admit a modification with right-continuous sample paths.