Skip to main content
SpringerLink
Log in

On the convergence of quadratic variation for compound fractional Poisson processes

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

The relationship between quadratic variation for compound renewal processes and M-Wright functions is discussed. The convergence of quadratic variation is investigated both as a random variable (for given t) and as a stochastic process.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Baleanu, K. Diethelm, E. Scalas, J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific & Imperial College Press, Singapore-London-etc. (2012); http://www.worldscibooks.com/mathematics/8180.html

    Book  MATH  Google Scholar 

  2. P. Becker-Kern, M.M. Meerschaert, and H. Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32, No 1B (2004), 730–756.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Beghin and E. Orshingher, Fractional Poisson processes and related planar random motions, Electr. J. Prob. 14, (2009), 1790–1826.

    MATH  Google Scholar 

  4. K. Bichteler, Stochastic integration and L p-theory of semimartingales, Ann. Probab. 9, No 1 (1981), 49–89.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Billingsley, Convergence of Probability Measures, Springer, New York (2002).

    Google Scholar 

  6. A. Erdélyi, W. Magnus, F. Oberhettinger, and F.-G. Tricomi, Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York (1955).

    Google Scholar 

  7. D. Fulger, E. Scalas, and G. Germano, Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation, Phys. Rev. E 77 (2008), 021122/1–7.

    Article  Google Scholar 

  8. G. Germano, M. Politi, E. Scalas, and R.L. Schilling, Stochastic calculus for uncoupled continuous-time random walks, Phys. Rev. E 79 (2009), 066102/1–12.

    Article  MathSciNet  Google Scholar 

  9. R. Gorenflo, Yu. Luchko and F. Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal. 2, No 4 (1999), 383–414; E-print at: http://arxiv.org/abs/math-ph/0701069

    MathSciNet  MATH  Google Scholar 

  10. R. Gorenflo, Yu. Luchko and F. Mainardi, Wright functions as scaleinvariant solutions of the diffusion-wave equation, J. Comp. Appl. Math. 118, No 1–2 (2000), 175–191.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto Fractional calculus and continuous-time finance III: The diffusion limit, In: M. Kohlmann and S. Tang (Editors), Mathematical Finance, Birkhäuser Verlag, Basel-Boston-Berlin (2001), 171–180.

    Chapter  Google Scholar 

  12. R. Gorenflo and F. Mainardi, Fractional diffusion processes: probability distributions and continuous time random walk, In: G. Rangarajan and M. Ding (Editors), Processes with Long Range Correlations, Springer-Verlag, Berlin (2003), 148–166 [Lecture Notes in Physics, No. 621]; E-print at http://arxiv.org/abs/0709.3990

    Chapter  Google Scholar 

  13. V. Kiryakova, The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Comput. Math. Appl. 59, No 3 (2010), 1128–1141; http://dx.doi.org/10.1016/j.camwa.2009.05.014

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Kiryakova, The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus, Comput. Math. Appl. 59, No 5 (2010), 1885–1895; http://dx.doi.org/10.1016/j.camwa.2009.08.025

    Article  MathSciNet  MATH  Google Scholar 

  15. A.N. Kolmogorov, On Skorohod convergence, Theory Probab. Appl. 1 (1979), 213–222.

    Google Scholar 

  16. N. Laskin, Fractional Poisson process, Commun. Nonlinear Science and Numerical Simulation 8 (2003), 201–213.

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, In: S. Rionero and T. Ruggeri (Editors), 7th Conf. on Waves and Stability in Continuous Media (WASCOM 1993), World Scientific, Singapore (1994), 246–251 [Ser. on Advances in Math. for Appl. Sci., Vol. 23].

    Google Scholar 

  18. F. Mainardi, Fractional relaxation-oscillation and fractional diffusionwave phenomena, Chaos, Solitons and Fractals 7, No 9 (1996), 1461–1477.

    Article  MathSciNet  MATH  Google Scholar 

  19. F. Mainardi, Yu. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal. 4, No 2 (2001), 153–192; E-print at http://arxiv.org/abs/cond-mat/0702419

    MathSciNet  MATH  Google Scholar 

  20. F. Mainardi, R. Gorenflo, and E. Scalas, A fractional generalization of the Poisson process, Vietnam J. Math. 32, SI (2004), 53–64; E-print at http://arxiv.org/abs/math/0701454

    MathSciNet  MATH  Google Scholar 

  21. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London (2010).

    Book  MATH  Google Scholar 

  22. F. Mainardi, A. Mura, and G. Pagnini, The M-Wright function in time-fractional diffusion processes: A tutorial survey, Intern. J. of Differential Equations 2010, Article ID 104505 (2010), 29 pages; E-print at http://arxiv.org/abs/1004.2950

    MathSciNet  Google Scholar 

  23. M.M. Meerschaert, E. Nane and P. Vellaisamy. The fractional Poisson process and the inverse stable subordinator. Electron. J. Probab. Vol. 16 (2011), no. 59, 1600–1620.

    MathSciNet  Google Scholar 

  24. M.M. Meerschaert, H.P. Scheffler, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. Wiley Series in Probability and Statistics (2001).

  25. M.M. Meerschaert, H.P. Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41, No. 3 (2004), 623–638.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Pagnini, Nonlinear time-fractional differential equations in combustion science, Fract. Calc. Appl. Anal. 14, No 1 (2011), 80–93; DOI: 10.2478/s13540-011-0006-8; at SpringerLink: http://www.springerlink.com/content/1311-0454/14/1/

    MathSciNet  Google Scholar 

  27. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999) [Mathematics in Science and Engineering, Vol. 198].

    MATH  Google Scholar 

  28. O.N. Repin and A.I. Saichev, Fractional Poisson law, Radiophysics and Quantum Electronics 43 No 9 (2000), 738–741.

    Article  MathSciNet  Google Scholar 

  29. K. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999).

    Google Scholar 

  30. E. Scalas, R. Gorenflo, F. Mainardi and M. Raberto, Revisiting the derivation of the fractional diffusion equation, Fractals 11S (2003), 281–289; E-print at http://arxiv.org/abs/cond-mat/0210166

    Article  MathSciNet  Google Scholar 

  31. E. Scalas, R. Gorenflo, and F. Mainardi, Uncoupled continuoustime random walks: Solution and limiting behavior of the master equation, Phys. Rev. E 69 (2004), 011107/1–8; E-print at http://arxiv.org/abs/cond-mat/0402657

    Article  MathSciNet  Google Scholar 

  32. E. Scalas, The application of continuous-time random walks in finance and economics, Physica A, 362 (2006), 225–239.

    Article  Google Scholar 

  33. A. V. Skorokhod, Limit theorems for stochastic processes. Theor. Probability Appl. 1 (1956), 261–290.

    Article  Google Scholar 

  34. C. Stone, Weak convergence of stochastic processes defined on semiinfinite time intervals, Proc. Amer. Math. Soc. 14, (1963), 694–696.

    Article  MathSciNet  MATH  Google Scholar 

  35. W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York (2002).

    Google Scholar 

  36. E.M. Wright, On the coefficients of power series having exponential singularities, Journal London Math. Soc. 8 (1933), 71–79.

    Article  Google Scholar 

  37. E.M. Wright, The asymptotic expansion of the generalized Bessel function, Proc. London Math. Soc. (Ser. II) 38 (1935), 257–270.

    Article  Google Scholar 

  38. E.M. Wright, The asymptotic expansion of the generalized hypergeometric function, Journal London Math. Soc. 10 (1935), 287–293.

    Article  Google Scholar 

  39. E.M. Wright, The generalized Bessel function of order greater than one, Quart. J. Math., Oxford Ser. 11 (1940), 36–48.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienze e Innovazione Tecnologica, Università del Piemonte Orientale, Viale T. Michel 11, I-15121, Alessandria, Italy

    Enrico Scalas

  2. BCAM-Basque Center of Applied Mathematics, Alameda de Mazarredo 14, E-48009, Bilbao, Basque Country, Spain

    Enrico Scalas & Noèlia Viles

Authors
  1. Enrico Scalas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Noèlia Viles
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Enrico Scalas.

About this article

Cite this article

Scalas, E., Viles, N. On the convergence of quadratic variation for compound fractional Poisson processes. fcaa 15, 314–331 (2012). https://doi.org/10.2478/s13540-012-0023-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s13540-012-0023-2

MSC 2010

Key Words and Phrases

Search

Navigation