 Solution

A pure birth process $N=(N(t):t\ge 0)$ is a continuous-time Markov chain on ${\mathbb{N}}_0$ whose only possible transition from state $n$ isEquivalently, after entering state $n$ it waits an independent exponential time $X_n\sim \mathrm{Exp}({\lambda }_n)$ and then moves to $n+1$.

The process is nonexplosive when it makes only finitely many jumps during every bounded time interval. If it starts from zero, this is equivalent to its explosion timebeing infinite almost surely.