Counting process
Encyclopedia
A counting process is a stochastic process
{N(t), t ≥ 0} that possesses the following properties:
If s < t, then N(t) − N(s) is the number of events occurred during the interval( s, t ] . 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 cummulative 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.
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
{N(t), t ≥ 0} that possesses the following properties:
- N(t) ≥ 0.
- N(t) is an integer.
- If s ≤ t then N(s) ≤ N(t).
If s < t, then N(t) − N(s) is the number of events occurred during the interval
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
and Renewal processes
Renewal theory
Renewal theory is the branch of probability theory that generalizes Poisson processes for arbitrary holding times. Applications include calculating the expected time for a monkey who is randomly tapping at a keyboard to type the word Macbeth and comparing the long-term benefits of different...
.
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 cummulative 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
Markov property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It was named after the Russian mathematician Andrey Markov....
, it is said to be a Markov counting process.