# counting process

A stochastic process $\{X(t)\mid t\in\mathbb{R}^{+}\cup\{0\}\}$ is called a counting process if, for each outcome $\omega$ in the sample space $\Omega$,

1. 1.

$X(t)\in\mathbb{Z}^{+}\cup\{0\}$ for all $t$,

2. 2.

$X(t)(\omega)$ is piecewise constant,

3. 3.

$X(t)(\omega)$ is non-decreasing,

4. 4.

$X(t)(\omega)$ is right continuous (continuous from the right), and

5. 5.

for any $t$, there is an $s\in\mathbb{R}$ such that $t and $X(t)(\omega)+1=X(s)(\omega)$.

Remark. For any $t$, the random variable $X(t)$ is usually called the number of occurrences of some event by time $t$. Then, for $s, $X(t)-X(s)$ is the number of occurrences in the half-open interval $(s,t]$.

Title counting process CountingProcess 2013-03-22 15:01:19 2013-03-22 15:01:19 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 60G51