We prove that a square-integrable set-indexed stochastic process is a set-indexed Brownian motion if and only if its projection on all the strictly increasing continuous sequences are one-parameter

The set-indexed Brownian motion

In this paper, we define a group action on the indexing collection

The frame of a set-indexed Brownian motion is not only a new step of generalization of a classical Brownian motion, but it proved a new look upon a Brownian motion. In recent years, there have been many new results related to the dynamical properties of random processes indexed by a class of sets. Set-indexed processes have many potential areas of applications. For example: environment (increased occurrence of polluted wells in a rural area could indicate a geographic region that has been subjected to industrial waste), astronomy (a cluster of black holes could be a result of an unobservable phenomenon affecting a region in space), quality control (an increased rate of breakdowns in a certain type of equipment might follow the failure of one or more components), population health (unusually frequent outbreaks of a disease such as leukemia near a nuclear power plant could signal a region of possible air or ground contamination), and the like.

Cairoli and Walsh [

In the last section, we present some results concerning the compensators of a set-indexed strong martingale and analyze the concept of path-independent variation in connection with independent increments in set-indexed process. We introduce compensators and demonstrate that the path-independent variation property permits a better understanding of the Doob–Meyer decomposition.

We recall the definitions and notation from [

Let (

(a) Note that any collection of sets closed under intersections is a semilattice with respect to the partial order of the inclusion.

(b) Definition

(a) The classical example is

(b) The example (a) may be generalized as follows. Let

Let

for all

for all

We will need other classes of sets generated by

In addition, let

Any

A set-indexed stochastic process

We shall always assume that our stochastic processes are additive. We note that a process with an (almost sure) additive extension to

Let

A positive measure

The classical examples for this definition are the Lebesgue measure or Radon measure when

Let

The classical examples are the following:

Let

Let

Let

The sequence

If

Given a set-indexed stochastic process

A set-indexed stochastic process

(a) It is clear from Lemma

(b) Notice that for each

Similarly to the construction performed in Lemma 2, we can prove that for all increasing sequences

Let

For any

(a) Let

(b) Let

The characterization of a set-indexed Brownian motion by a group action on a sequence (Theorem

(

(

(

(

Let

A

A martingale if for any

For studies of different kinds of martingales, see [

In particular, if

Under some hypotheses, we can define

From the well-known Lévy martingale characterization of the Brownian motion (see [

(

(

In addition, Theorem

Let

write

write

write

Let

if

if

Let

We must show that if

Let

It is clear that

Clearly, we have

Let

Let

Let

Let

Let

(The set

From Theorem

Let

Hereafter, we assume that the space

The classical example for definition is

Let

Let

A square-integrable set-indexed martingale

(a) This definition of s.i.v. on

(b) The definition and more details about

It suffices to prove that for all increasing continuous sequences

For the proof, we need two auxiliary propositions.

The process

Now, for any increasing continuous sequences

It is clear that

Let

Since

Let

Returning to the proof of Theorem

Now, if