Compound Poisson Process with a Poisson Subordinator. (June 2015)

• Record Type: Journal Article
• Authors: Di Crescenzo, Antonio; Martinucci, Barbara; Zacks, Shelemyahu
• Is Part Of: Journal of applied probability. Volume 52:Number 2(2015)
• Page Start: 360
• Page End: 374
• Publication Date: 2015-06
• Subjects: Bell polynomials, -- first-crossing time, -- iterated process, -- linear boundary, -- mean sojourn time, -- Poisson process; 60J27, -- 60G40; 519.2
• DOI: 10.1017/S0021900200012511
• Languages: English
• ISSNs: 0021-9002

Abstract

Abstract : 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.