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Multivariate fractional Poisson processes and compound sums

Part of: Real functions Other special functions Stochastic processes

Published online by Cambridge University Press:  19 September 2016

Luisa Beghin  and
Claudio Macci
Luisa Beghin*
Affiliation:
Sapienza Università di Roma
Claudio Macci*
Affiliation:
Università di Roma Tor Vergata
*
* Postal address: Dipartimento di Scienze Statistiche, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy. Email address: luisa.beghin@uniroma1.it
** Postal address: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy. Email address: macci@mat.uniroma2.it
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Abstract

In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (nonfractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.

Keywords

Conditional independenceFox‒Wright functionfractional differential equationrandom time-change

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion 60G52: Stable processes
Secondary: 26A33: Fractional derivatives and integrals 33E12: Mittag-Leffler functions and generalizations

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 691 - 711
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Applebaum, D. (2009).Lévy Processes and Stochastic Calculus,2nd edn.Cambridge University Press.CrossRefGoogle Scholar
[2] Beghin, L. and D'Ovidio, M. (2014).Fractional Poisson process with random drift.Electron. J. Prob. 19, 26pp.Google Scholar
[3] Beghin, L. and Macci, C. (2014).Fractional discrete processes: compound and mixed Poisson representations.J. Appl. Prob. 51,19–36.Google Scholar
[4] Beghin, L. and Orsingher, E. (2009).Fractional Poisson processes and related planar motions.Electron. J. Prob. 14,1790–1827.Google Scholar
[5] Beghin, L. and Orsingher, E. (2010).Poisson-type processes governed by fractional and higher-order recursive differential equations.Electron. J. Prob. 15,684–709.CrossRefGoogle Scholar
[6] Biard, R. and Saussereau, B. (2014).Fractional Poisson process: long-range dependence and applications in ruin theory.J. Appl. Prob. 51,727–740.Google Scholar
[7] Hahn, M. G.,Kobayashi, K. and Umarov, S. (2011).Fokker‒Planck‒Kolmogorov equations associated with time-changed fractional Brownian motion.Proc. Amer. Math. Soc. 139,691–705.Google Scholar
[8] Kilbas, A. A.,Srivastava, H. M. and Trujillo, J. J. (2006).Theory and Applications of Fractional Differential Equations.Elsevier,Amsterdam.Google Scholar
[9] Kokoszka, P. S. and Taqqu, M. S. (1996).Infinite variance stable moving averages with long memory.J. Econometrics 73,79–99.CrossRefGoogle Scholar
[10] Kumar, A.,Nane, E. and Vellaisamy, P. (2011).Time-changed Poisson processes.Statist. Prob. Lett. 81,1899–1910.Google Scholar
[11] Laskin, N. (2003).Fractional Poisson process.Commun. Nonlinear Sci. Numer. Simul. 8,201–213.Google Scholar
[12] Mainardi, F.,Gorenflo, R. and Scalas, E. (2004).A fractional generalization of the Poisson process.Vietnam J. Math. 32,53–64.Google Scholar
[13] Meerschaert, M. M.,Nane, E. and Vellaisamy, P. (2011).The fractional Poisson process and the inverse stable subordinator.Electron. J. Prob. 16,1600–1620.CrossRefGoogle Scholar
[14] Minkova, L. D. (2004).The Pólya‒Aeppli process and ruin problems.J. Appl. Math. Stoch. Analysis 2004,221–234.Google Scholar
[15] Orsingher, E. and Polito, F. (2012).The space-fractional Poisson process.Statist. Prob. Lett. 82,852–858.Google Scholar
[16] Orsingher, E. and Toaldo, B. (2015).Counting processes with Bernštein intertimes and random jumps.J. Appl. Prob. 52,1028–1044.Google Scholar
[17] Piryatinska, A.,Saichev, A. I. and Woyczynski, W. A. (2005).Models of anomalous diffusion: the subdiffusive case.Physica A 349,375–420.CrossRefGoogle Scholar
[18] Podlubny, I. (1999).Fractional Differential Equations.Academic Press,San Diego, CA.Google Scholar
[19] Politi, M.,Kaizoji, T. and Scalas, E. (2011).Full characterization of the fractional Poisson process.Europhys. Lett. 96, 20004.Google Scholar
[20] Repin, O. N. and Saichev, A. I. (2000).Fractional Poisson law.Radiophys. Quantum Electron. 43,738–741.Google Scholar
[21] Sato, K.-I. (1999).Lévy Processes and Infinitely Divisible Distributions.Cambridge University Press.Google Scholar
[22] Scalas, E. and Viles, N. (2012).On the convergence of quadratic variation for compound fractional Poisson processes.Fract. Calc. Appl. Analysis 15,314–331.Google Scholar
[23] Srivastava, R. (2013).Some generalizations of Pochhammer's symbol and their associated families of hypergeometric functions and hypergeometric polynomials.Appl. Math. Inf. Sci. 7,2195–2206.CrossRefGoogle Scholar