counting process

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

1.
$X(t)\\in\\mathbb{Z}^{+}\\cup\\{0\\}$ for all $t$ ,
2.
$X(t)(\\omega)$ is piecewise constant,
3.
$X(t)(\\omega)$ is non-decreasing,
4.
$X(t)(\\omega)$ is right continuous (continuous from the right), and
5.
for any $t$ , there is an $s\\in\\mathbb{R}$ such that $t<s$ 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 $s<t$ , $X(t)-X(s)$ is the number of occurrences in the half-open interval $(s,t]$ .

单词	CountingProcess
释义	counting process A stochastic process $\\{X(t)\\mid t\\in\\mathbb{R}^{+}\\cup\\{0\\}\\}$ is called acounting process if, for each outcome $\\omega$ in the sample space $\\Omega$ , 1. $X(t)\\in\\mathbb{Z}^{+}\\cup\\{0\\}$ for all $t$ , 2. $X(t)(\\omega)$ is piecewise constant, 3. $X(t)(\\omega)$ is non-decreasing, 4. $X(t)(\\omega)$ is right continuous (continuous from the right), and 5. for any $t$ , there is an $s\\in\\mathbb{R}$ such that $t<s$ 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 $s<t$ , $X(t)-X(s)$ is the number of occurrences in the half-open interval $(s,t]$ .
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