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=AB, where A𝒜 and B𝒜(u) (finite unions of sets from 𝒜). Moreover, for any C=AB, B= i=1 k A i , 𝒜 𝐂 is defined as the following subset of 𝒜:

𝒜 𝐂 ={U𝒜 ;UB }:={U C 1 ,,U C p },wherep=|𝒜 𝐂 |

and 𝒜 corresponds to the semilattice {A 1 A k ,,A 1 A 2 ,A 1 ,A k }𝒜. The notation 𝐗 𝐂 refers to a random vector 𝐗 𝐂 =X U C 1 , , X U C p . Similarly, 𝐱 𝐂 is used to denote a vector of variables (x U C 1 ,,x U C p ).

Then, an E-valued set-indexed process (X A ) A𝒜 is said to be 𝒞-Markov with respect to a filtration ( A ) A𝒜 if it is adapted to ( A ) A𝒜 and if it satisfies

𝔼[f(X A )|𝒢 C * ]=𝔼[f(X A )|𝐗 𝐂 ]-a.s.(7)

for all C=AB𝒞 and any bounded measurable function f:E𝐑. The sigma-algebra 𝒢 C * is usually called the strong history of ( A ) A𝒜 and is defined as 𝒢 C * = A𝒜,AC= 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 ;dx A );C𝒞} as

𝐱 𝐂 E |𝒜 𝐂 | ,Γ;P C (𝐱 𝐂 ;Γ):=(X A Γ|𝐗 𝐂 =𝐱 𝐂 ).

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

C𝒞,A ' 𝒜;P C f=P C ' P C '' fwhereC ' =CA ' ,C '' =CA ' (8)

and f is a bounded measurable function.

Similarly to the classic Markovian theory, is is proved in [1] that the initial distribution μ 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

C=AB𝒞,Γ;P C (𝐱 𝐂 ;Γ)=μ m(C) (Γ-Δx B ),(9)

where m is a measure on 𝒯 and μ 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 C (𝐱 𝐂 ;y)=1 σ C 2πexp- 1 2σ C 2 y - e -λm(A) i=1 n (-1) ε i x U C i e λm(U C i ) 2 ,(10)

where λ and σ are positive parameters, m is a measure on 𝒯 and

σ C 2 =σ 2 2λ1 - e -2λm(A) i=1 n (-1) ε i e 2λm(U C i ) .

In the particular case of multiparameter processes, corresponding to the indexing collection 𝒜={[0,t];t𝐑 + 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.