Advances in Applied Probability

Multivariate fractional Poisson processes and compound sums

Luisa Beghin and Claudio Macci

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Adv. in Appl. Probab., Volume 48, Number 3 (2016), 691-711.

Dates
First available in Project Euclid: 19 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1474296310

Mathematical Reviews number (MathSciNet)
MR3568887

Zentralblatt MATH identifier
1351.60045

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

Keywords
Conditional independence Fox‒Wright function fractional differential equation random time-change

Citation

Beghin, Luisa; Macci, Claudio. Multivariate fractional Poisson processes and compound sums. Adv. in Appl. Probab. 48 (2016), no. 3, 691--711. https://projecteuclid.org/euclid.aap/1474296310


Export citation

References

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