counting process

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

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 $s\in ℝ$ such that 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 ,$X(t)-X(s)$ is the number of occurrences in the half-open interval$(s,t]$.