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Compound Poisson Process with a Poisson Subordinator

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Antonio Di Crescenzo ,
Barbara Martinucci  and
Shelemyahu Zacks
Antonio Di Crescenzo*
Affiliation:
Università degli Studi di Salerno
Barbara Martinucci*
Affiliation:
Università degli Studi di Salerno
Shelemyahu Zacks*
Affiliation:
Binghamton University
*
∗ Postal address: Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, n. 132, 84084 Fisciano (SA), Italy.
∗ Postal address: Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, n. 132, 84084 Fisciano (SA), Italy.
∗∗∗∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA. Email address: shelly@math.binghamton.edu
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Abstract

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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.

Keywords

Bell polynomialsfirst-crossing timeiterated processlinear boundarymean sojourn timePoisson process

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 2 , June 2015 , pp. 360 - 374
Copyright
© Applied Probability Trust 

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