A non-homogeneous Poisson Process is parameterized by its intensity (or rate) function $r(t)$ for $t \in [0, \infty)$. Often what is assumed in the literature about the function $r$ is that $$\int_0^{\infty}r(t)dt = \infty.$$ I understand that this assumption ensures that with probability 1 the process has a jump after any time point $t$. My question is, besides this implication, what would be missing without this unbounded integral assumption on $r$. For instance, if $r$ has only finite support over a closed interval $[0, T]$, is there any reason that one should try to avoid using such a function as the rate function of a Poisson process? Thanks!
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No this is fine. Such processes occur frequently. There are some technicalities, but it will simply define a process which simply consists of a finite number of points on $[0,T]$. – Pax Jul 08 '21 at 22:37
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@max - Thanks, max! Not sure if a comment is appropriate for a follow-up question: taking the associated probability density function $f$ of the first arrival time, what are the implications on $f$ if $r$ has bounded support? Do I need that the integral of $f$ from 0 to $T$ to be precisely 1? – Xudong Zheng Jul 09 '21 at 00:49