Compound Poisson process with a Poisson subordinator

TL;DR

This paper analyzes a compound Poisson process with a Poisson subordinator, deriving its distribution, exploring special jump cases, and studying properties including convergence and first-crossing-time problems.

Contribution

It provides a new probability distribution for the process using Bell polynomials and investigates special jump cases and crossing-time properties.

Findings

Distribution expressed via Bell polynomials

Convergence to a Poisson process established

Closed-form and iterative solutions for crossing times

Abstract

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.