Electronic Journal of Probability

Fractional Poisson process with random drift

Luisa Beghin and Mirko D'Ovidio

Full-text: Open access

Abstract

We study the connection between PDEs and Lévy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenius-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 122, 26 pp.

Dates
Accepted: 30 December 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065764

Digital Object Identifier
doi:10.1214/EJP.v19-3258

Mathematical Reviews number (MathSciNet)
MR3304182

Zentralblatt MATH identifier
1334.60053

Subjects
Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]
Secondary: 60G50: Sums of independent random variables; random walks

Keywords
Poisson process time-change random drift fractional equation Poisson semigroup

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Beghin, Luisa; D'Ovidio, Mirko. Fractional Poisson process with random drift. Electron. J. Probab. 19 (2014), paper no. 122, 26 pp. doi:10.1214/EJP.v19-3258. https://projecteuclid.org/euclid.ejp/1465065764


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References

  • Applebaum, David. Lévy processes and stochastic calculus. Cambridge Studies in Advanced Mathematics, 93. Cambridge University Press, Cambridge, 2004. xxiv+384 pp. ISBN: 0-521-83263-2
  • Beghin, L.; Orsingher, E. Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14 (2009), no. 61, 1790–1827.
  • Beghin, L.; Orsingher, E. Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Probab. 15 (2010), no. 22, 684–709.
  • D'Ovidio, Mirko. On the fractional counterpart of the higher-order equations. Statist. Probab. Lett. 81 (2011), no. 12, 1929–1939.
  • M. D'Ovidio. Wright functions governed by fractional directional derivatives and fractional advection diffusion equations. phMethods and Applications of Analysis, to appear.
  • W. Feller. An introduction to probability theory and its applications, v.2. Wiley, New York, 2nd ed., 1971.
  • Forst, Gunnar. Subordinates of the Poisson semigroup. Z. Wahrsch. Verw. Gebiete 55 (1981), no. 1, 35–40.
  • Doetsch, Gustav. Introduction to the theory and application of the Laplace transformation. Translated from the second German edition by Walter Nader. Springer-Verlag, New York-Heidelberg, 1974. vii+326 pp.
  • Germano, Guido; Politi, Mauro; Scalas, Enrico; Schilling, René L. Stochastic calculus for uncoupled continuous-time random walks. Phys. Rev. E (3) 79 (2009), no. 6, 066102, 12 pp.
  • Hille, Einar; Phillips, Ralph S. Functional analysis and semi-groups. rev. ed. American Mathematical Society Colloquium Publications, vol. 31. American Mathematical Society, Providence, R. I., 1957. xii+808 pp.
  • James, Lancelot F. Lamperti-type laws. Ann. Appl. Probab. 20 (2010), no. 4, 1303–1340.
  • Laskin, Nick. Fractional Poisson process. Chaotic transport and complexity in classical and quantum dynamics. Commun. Nonlinear Sci. Numer. Simul. 8 (2003), no. 3-4, 201–213.
  • Kilbas, Anatoly A.; Srivastava, Hari M.; Trujillo, Juan J. Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. xvi+523 pp. ISBN: 978-0-444-51832-3; 0-444-51832-0
  • Meerschaert, Mark M.; Nane, Erkan; Vellaisamy, P. The fractional Poisson process and the inverse stable subordinator. Electron. J. Probab. 16 (2011), no. 59, 1600–1620.
  • Mainardi, Francesco; Gorenflo, Rudolf; Scalas, Enrico. A fractional generalization of the Poisson processes. Vietnam J. Math. 32 (2004), Special Issue, 53–64.
  • Orsingher, Enzo; Polito, Federico. The space-fractional Poisson process. Statist. Probab. Lett. 82 (2012), no. 4, 852–858.
  • Repin, O. N.; Saichev, A. I. Fractional Poisson law. Radiophys. and Quantum Electronics 43 (2000), no. 9, 738–741 (2001).
  • Scalas, Enrico; Viles, Noèlia. On the convergence of quadratic variation for compound fractional Poisson processes. Fract. Calc. Appl. Anal. 15 (2012), no. 2, 314–331.
  • Schilling, René L.; Song, Renming; Vondracek, Zoran. Bernstein functions. Theory and applications. Second edition. de Gruyter Studies in Mathematics, 37. Walter de Gruyter & Co., Berlin, 2012. xiv+410 pp. ISBN: 978-3-11-025229-3; 978-3-11-026933-8
  • Traple, Janusz. Markov semigroups generated by Poisson driven differential equations. Bull. Polish Acad. Sci. Math. 44 (1996), no. 2, 161–182.