Skip to main content Accessibility help
×
Home

Compound Poisson Process with a Poisson Subordinator

  • Antonio Di Crescenzo (a1), Barbara Martinucci (a1) and Shelemyahu Zacks (a2)

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.

Copyright

Corresponding author

Postal address: Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, n. 132, 84084 Fisciano (SA), Italy.
∗∗ Email address: adicrescenzo@unisa.it
∗∗∗ Email address: bmartinucci@unisa.it
∗∗∗∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA. Email address: shelly@math.binghamton.edu

References

Hide All
[1] Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press.
[2] Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Prob. 14, 17901827.
[3] Beghin, L. and Orsingher, E. (2010). Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Prob. 15, 684709.
[4] Beghin, L. and Orsingher, E. (2012). Poisson process with different Brownian clocks. Stochastics 84, 79112.
[5] Comtet, L. (1974). Advanced Combinatorics. The Art of Finite and Infinite Expansions. Reidel, Dordrecht.
[6] Di Crescenzo, A. and Martinucci, B. (2009). On a first-passage-time problem for the compound power-law process. Stoch. Models 25, 420435.
[7] Grandell, J. (1976). Doubly Stochastic Poisson Processes (Lecture Notes Math. 529). Springer, Berlin.
[8] Horváth, L. and Steinebach, J. (1999). On the best approximation for bootstrapped empirical processes. Statist. Prob. Lett. 41, 117122.
[9] Kumar, A., Nane, E. and Vellaisamy, P. (2011). Time-changed Poisson processes. Statist. Prob. Lett. 81, 18991910.
[10] Lee, M. L. T. and Whitmore, G. A. (1993). Stochastic processes directed by randomized time. J. Appl. Prob. 30, 302314.
[11] Mainardi, F., Gorenflo, F. and Scalas, E. (2004). A fractional generalization of the Poisson processes. Vietnam J. Math. 32, 5364.
[12] Orsingher, E. and Polito, F. (2010). Composition of Poisson processes. In Proceedings of XIV International Conference on Eventological Mathematics and Related Fields (Krasnoyarsk, Russia), pp. 1318.
[13] Orsingher, E. and Polito, F. (2012). Compositions, random sums and continued random fractions of Poisson and fractional Poisson processes. J. Statist. Phys. 148, 233249.
[14] Orsingher, E. and Toaldo, B. (2015). Counting processes with Bernštein intertimes and random Jumps. To appear in J. Appl. Prob. 52.
[15] Pickands, J. III (1971). The two-dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745756.
[16] Serfozo, R. F. (1972). Conditional Poisson processes. J. Appl. Prob. 9, 288302.
[17] Stadje, W. and Zacks, S. (2003). Upper first-exit times of compound Poisson processes revisited. Prob. Eng. Inf. Sci. 17, 459465.
[18] Zacks, S. (1991). Distributions of stopping times for Poisson processes with linear boundaries. Commun. Statist. Stoch. Models 7, 233242.
[19] Zacks, S. (2005). Some recent results on the distributions of stopping times of compound Poisson processes with linear boundaries. J. Statist. Planning Infer. 130, 95109.

Keywords

MSC classification

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed