# Counting process

A counting process is a stochastic process {N(t), t ≥ 0} with values that are positive, integer, and increasing:

1. N(t) ≥ 0.
2. N(t) is an integer.
3. If st then N(s) ≤ N(t).

If s < t, then N(t) − N(s) is the number of events occurred during the interval [st ]. Examples of counting processes include Poisson processes and Renewal processes.

Because of the third property, a counting process is increasing and hence a submartingale. Then by Doob-Meyer, it can be written as

$N(t)=M(t)+A(t)$

with a martingale M(t) and a predictable increasing process A(t). The martingale M(t) is called the martingale associated with the counting process N(t) and the predictable process A(t) is called the cumulative intensity of the counting process N(t).

Counting processes deal with the number of various outcomes in a system over time. An example of a counting process is the number of occurrences of "heads" over some number of coin tosses.

If a process has the Markov property, it is said to be a Markov counting process.

## §References

• Ross, S.M. (1995) Stochastic Processes. Wiley. ISBN 978-0-471-12062-9
• Higgins JJ, Keller-McNulty S (1995) Concepts in Probability and Stochastic Modeling. Wadsworth Publishing Company. ISBN 0-534-23136-5