Let $N(t)$ be a doubly stochastic Poisson process with stochastic intensity $\lambda(t)$. Moreover, let us assume that the intensity is lower bounded by $\bar{\lambda}>0:$ $$\lambda(t)>\bar{\lambda}$$ for all $t>0.$
How can I prove that the process tends to infinity almost surely: $$N(t)\rightarrow \infty \quad a.s.$$ as $t\rightarrow \infty$ ?
This result is well-known for a Poisson process with constant intensity but I can't find a similar result for stochastic intensity. Let us remark that I made the assumption of a strictly positive lower bound to avoid the situation of a Poisson process with vanishing intensity.
Thanks for your help !
In P. Brémaud's book, Point Process Calculus in Time and Space, Springer Natural Switzerland AG 2020, Th5.1.16, p.171, there is following theorem:
Theorem 5.1.16 Let $N$ be a simple locally finite point process on $\mathbb{R}_+ $ with the $\mathscr{F}_t $-intensity $\{\lambda(t)\}_{t\ge 0}$. Then \begin{equation*} N(\infty)<\infty \iff \int_0^\infty \lambda(s)\,\mathrm{d}s<\infty, \quad P-a.s. \end{equation*} This result is sufficient for your require.