be some measurable space
. The Poisson random measure
with intensity measure
is a family of random variables
defined on some probability space
i) is a Poisson random variable with rate .
ii) If sets don't intersect then the corresponding random variables from i) are mutually independent.
iii) is a measure on
satisfies the conditions i)-iii). Otherwise, in the case of finite measure
- Poisson random variable
- mutually independent random variables
is a degenerate measure
will be a Poisson random measure. In the case
is not finite the measure
can be obtained from the measures constructed above on parts of
This kind of random measure
is often used when describing jumps of stochastic processes
, in particular in Lévy-Itō decomposition
of the Lévy processes
- Sato K. Lévy Processes and Infinitely Divisible Distributions Cambridge University Press, (1st ed.) ISBN 0-521-55302-4.