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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 $𝒜$:

and ${𝒜}_{\ell }$ corresponds to the semilattice $\{A_1\cap \cdots \cap A_k,\cdots ,A_1\cap A_2,A_1\cdots ,A_k\}\subset 𝒜$. The notation ${𝐗}_{𝐂}$ refers to a random vector ${𝐗}_{𝐂}=\left(X_{U_C^1},\cdots ,X_{U_C^p}\right)$. Similarly, ${𝐱}_{𝐂}$ is used to denote a vector of variables $(x_{U_C^1},\cdots ,x_{U_C^p})$.

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

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

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 $𝒫=\{P_C({𝐱}_{𝐂};\mathrm{d}{x}_A);C\in 𝒞\}$ as

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

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

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

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

In the particular case of multiparameter processes, corresponding to the indexing collection $𝒜=\{[0,t];t\in {𝐑}_{+}^N\}$, 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.