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Multivariate fractional Poisson processes and compound sums

  • Luisa Beghin (a1) and Claudio Macci (a2)

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.

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Corresponding author

* 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

References

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[1] Applebaum, D. (2009).Lévy Processes and Stochastic Calculus,2nd edn.Cambridge University Press.
[2] Beghin, L. and D'Ovidio, M. (2014).Fractional Poisson process with random drift.Electron. J. Prob. 19, 26pp.
[3] Beghin, L. and Macci, C. (2014).Fractional discrete processes: compound and mixed Poisson representations.J. Appl. Prob. 51,1936.
[4] Beghin, L. and Orsingher, E. (2009).Fractional Poisson processes and related planar motions.Electron. J. Prob. 14,17901827.
[5] Beghin, L. and Orsingher, E. (2010).Poisson-type processes governed by fractional and higher-order recursive differential equations.Electron. J. Prob. 15,684709.
[6] Biard, R. and Saussereau, B. (2014).Fractional Poisson process: long-range dependence and applications in ruin theory.J. Appl. Prob. 51,727740.
[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. 139,691705.
[8] Kilbas, A. A.,Srivastava, H. M. and Trujillo, J. J. (2006).Theory and Applications of Fractional Differential Equations.Elsevier,Amsterdam.
[9] Kokoszka, P. S. and Taqqu, M. S. (1996).Infinite variance stable moving averages with long memory.J. Econometrics 73,7999.
[10] Kumar, A.,Nane, E. and Vellaisamy, P. (2011).Time-changed Poisson processes.Statist. Prob. Lett. 81,18991910.
[11] Laskin, N. (2003).Fractional Poisson process.Commun. Nonlinear Sci. Numer. Simul. 8,201213.
[12] Mainardi, F.,Gorenflo, R. and Scalas, E. (2004).A fractional generalization of the Poisson process.Vietnam J. Math. 32,5364.
[13] Meerschaert, M. M.,Nane, E. and Vellaisamy, P. (2011).The fractional Poisson process and the inverse stable subordinator.Electron. J. Prob. 16,16001620.
[14] Minkova, L. D. (2004).The Pólya‒Aeppli process and ruin problems.J. Appl. Math. Stoch. Analysis 2004,221234.
[15] Orsingher, E. and Polito, F. (2012).The space-fractional Poisson process.Statist. Prob. Lett. 82,852858.
[16] Orsingher, E. and Toaldo, B. (2015).Counting processes with Bernštein intertimes and random jumps.J. Appl. Prob. 52,10281044.
[17] Piryatinska, A.,Saichev, A. I. and Woyczynski, W. A. (2005).Models of anomalous diffusion: the subdiffusive case.Physica A 349,375420.
[18] Podlubny, I. (1999).Fractional Differential Equations.Academic Press,San Diego, CA.
[19] Politi, M.,Kaizoji, T. and Scalas, E. (2011).Full characterization of the fractional Poisson process.Europhys. Lett. 96, 20004.
[20] Repin, O. N. and Saichev, A. I. (2000).Fractional Poisson law.Radiophys. Quantum Electron. 43,738741.
[21] Sato, K.-I. (1999).Lévy Processes and Infinitely Divisible Distributions.Cambridge University Press.
[22] Scalas, E. and Viles, N. (2012).On the convergence of quadratic variation for compound fractional Poisson processes.Fract. Calc. Appl. Analysis 15,314331.
[23] Srivastava, R. (2013).Some generalizations of Pochhammer's symbol and their associated families of hypergeometric functions and hypergeometric polynomials.Appl. Math. Inf. Sci. 7,21952206.

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