  A compound Poisson process whose randomized time is an independent Poisson
process is called compound Poisson process with Poisson subordinator. We
provide its probability distribution, which is expressed in terms of the Bell
polynomials, and investigate in detail both the special cases in which the
compound Poisson process has exponential jumps and normal jumps. Then for the
iterated Poisson process we discuss some properties and provide convergence
results to a Poisson process. The first-crossing-time problem for the iterated
Poisson process is finally tackled in the cases of (i) a decreasing and
constant boundary, where we provide some closed-form results, and (ii) a
linearly increasing boundary, where we propose an iterative procedure to
compute the first-crossing-time density and survival functions.
