Point processes subordinated to compound Poisson processes
Authors:
K. V. Kobylych and L. M. Sakhno
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 94 (2017), 89-96
MSC (2010):
Primary 60G55, 60G50
DOI:
https://doi.org/10.1090/tpms/1011
Published electronically:
August 25, 2017
MathSciNet review:
3553456
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Abstract: Point processes $N^f (t)=N\bigl (H^f(t)\bigr )$, $t>0$, are studied in the paper where $N(t)$ is a Poisson process and $H^f(t)$ is a subordinator with the Berns̆tein function $f(\lambda )$. We present the probability distribution and moments of the first and second order of processes $N^f(t)$ for the case where $H^f(t)$ is a compound Poisson process with gamma distributed jumps. We also consider these processes with double and iterated time change.
References
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References
- D. Applebaum, Lévy Processes and Stochastic Calculus, second edition, Cambridge University Press, Cambridge, 2009. MR 2512800
- L. Beghin and C. Macci, Alternative forms of compound fractional Poisson processes, Abstract Appl. Anal., 2012 (2012), article ID 747503, 30 pages. MR 2991021
- R. Mendoza-Arriaga and V. Linetsky, Time-changed CIR default intensities with two-sided mean-reverting jumps, Ann. Appl. Probab. 24 (2014), no. 2, 811–856. MR 3178498
- R. Garra, E. Orsingher, and M. Scavino, Some probabilistic properties of fractional point processes, Stoch. Anal. Appl. 35 (2017), no. 4, 701–718. MR 3651139
- E. Orsingher and F. Polito, The space-fractional Poisson process, Stat. Probab. Letters 82 (2012), 852–858. MR 2899530
- E. Orsingher and B. Toaldo, Counting processes with Berns̆tein intertimes and random jumps, J. Appl. Probab. 52 (2015), 1028–1044. MR 3439170
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Additional Information
K. V. Kobylych
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 4E, Kyiv 03127, Ukraine
Email:
kristina.kobilich@gmail.com
L. M. Sakhno
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 4E, Kyiv 03127, Ukraine
Email:
lms@univ.kiev.ua
Keywords:
Point processes,
Poisson processes,
generalized Poisson processes,
Berns̆tein function,
subordinators
Received by editor(s):
April 4, 2016
Published electronically:
August 25, 2017
Article copyright:
© Copyright 2017
American Mathematical Society