(Poisson point process) Let
be a filtered probability space,
be a measurable space and
,
.
A
-valued
process
is called "Poisson point process" if it has the following properties.

1. The map
is
- measurable.

2. The set
is countable a.s.

3. There is a sequence of sets
s.t.
and
where

4. The process
is
-adapted.

5. For any
,
and any
the distribution of
conditioned on
is the same as the distribution of
.

Proposition

(Independence of Poisson
processes) Two Poisson processes adapted to the same filtration are
independent iff they almost never jump simultaneously.

Proposition

(Properties of Poisson point
process) Let
be a Poisson point process.

1.
is a Poisson process for any
.

2. For
s.t.
the processes
and
are independent.

3. The quantity
is independent of
and constitutes a
-finite
measure on
.

Definition

(Characteristic
measure of Poisson point process) The measure
defined in the proposition
(
Properties of Poisson point
process
) is called "characteristic measure of Poisson point process".