Journal of Applied Probability

Compound Poisson process with a Poisson subordinator

Antonio Di Crescenzo, Barbara Martinucci, and Shelemyahu Zacks

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A compound Poisson process whose randomized time is an independent Poisson process is called a 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.

Article information

J. Appl. Probab. Volume 52, Number 2 (2015), 360-374.

First available in Project Euclid: 23 July 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Bell polynomials first-crossing time iterated process linear boundary mean sojourn time Poisson process


Di Crescenzo, Antonio; Martinucci, Barbara; Zacks, Shelemyahu. Compound Poisson process with a Poisson subordinator. J. Appl. Probab. 52 (2015), no. 2, 360--374. doi:10.1239/jap/1437658603.

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