Electronic Communications in Probability

Geometric stable processes and related fractional differential equations

Luisa Beghin

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Abstract

We are interested in the differential equations satisfied by the density of the Geometric Stable processes $\mathcal{G}_{\alpha }^{\beta }=\left\{\mathcal{G}_{\alpha }^{\beta }(t);t\geq 0\right\} $, with stability \ index $\alpha \in (0,2]$ and symmetry parameter $\beta \in \lbrack -1,1]$, both in the univariate and in the multivariate cases. We resort to their representation as compositions of stable processes with an independent Gamma subordinator. As a preliminary result, we prove that the latter is governed by a differential equation expressed by means of the shift operator. As a consequence, we obtain the space-fractional equation satisfied by the density of $\mathcal{G}_{\alpha }^{\beta }$. For some particular values of $\alpha $ and $\beta $, we get some interesting results linked to well-known processes, such as the Variance Gamma process and the first passage time of the Brownian motion.

Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 13, 14 pp.

Dates
Accepted: 1 March 2014
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465316715

Digital Object Identifier
doi:10.1214/ECP.v19-2771

Mathematical Reviews number (MathSciNet)
MR3174831

Zentralblatt MATH identifier
1321.60101

Subjects
Primary: 60G52: Stable processes
Secondary: 34A08: Fractional differential equations 33E12: Mittag-Leffler functions and generalizations 26A33: Fractional derivatives and integrals

Keywords
Symmetric Geometric Stable law Geometric Stable subordinator Shift operator Riesz-Feller fractional derivative Gamma subordinator

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

Citation

Beghin, Luisa. Geometric stable processes and related fractional differential equations. Electron. Commun. Probab. 19 (2014), paper no. 13, 14 pp. doi:10.1214/ECP.v19-2771. https://projecteuclid.org/euclid.ecp/1465316715


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References

  • Anderson, Dale N. A multivariate Linnik distribution. Statist. Probab. Lett. 14 (1992), no. 4, 333–336.
  • Anderson, Dale N.; Arnold, Barry C. Linnik distributions and processes. J. Appl. Probab. 30 (1993), no. 2, 330–340.
  • Applebaum, David. Lévy processes and stochastic calculus. Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. xxx+460 pp. ISBN: 978-0-521-73865-1
  • Balakrishnan, A. V. Fractional powers of closed operators and the semigroups generated by them. Pacific J. Math. 10 1960 419–437.
  • Bochner, S. Diffusion equation and stochastic processes. Proc. Nat. Acad. Sci. U. S. A. 35, (1949). 368–370.
  • Bogdan, Krzysztof; Byczkowski, Tomasz; Kulczycki, Tadeusz; Ryznar, Michal; Song, Renming; Vondraček, Zoran. Potential analysis of stable processes and its extensions. Edited by Piotr Graczyk and Andrzej Stos. Lecture Notes in Mathematics, 1980. Springer-Verlag, Berlin, 2009. x+187 pp. ISBN: 978-3-642-02140-4
  • Doney, R. A. On Wiener-Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Probab. 15 (1987), no. 4, 1352–1362.
  • ErdoÄŸan, M. Burak. Analytic and asymptotic properties of non-symmetric Linnik's probability densities. J. Fourier Anal. Appl. 5 (1999), no. 6, 523–544.
  • Grzywny, Tomasz; Ryznar, Michał. Potential theory of one-dimensional geometric stable processes. Colloq. Math. 129 (2012), no. 1, 7–40.
  • Jayakumar, K., Suresh, R.P.: Mittag-Leffler distributions. phJ. Indian Soc. Probab. Statist. 7, (2003), 51–71.
  • 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
  • Kotz, Samuel; Kozubowski, Tomasz J.; Podgorski, Krzysztof. The Laplace distribution and generalizations. A revisit with applications to communications, economics, engineering, and finance. Birkhäuser Boston, Inc., Boston, MA, 2001. xviii+349 pp. ISBN: 0-8176-4166-1
  • Kozubowski, T. J.; Rachev, S. T. Univariate geometric stable laws. J. Comput. Anal. Appl. 1 (1999), no. 2, 177–217.
  • Kozubowski, T. J. Fractional moment estimation of Linnik and Mittag-Leffler parameters. Stable non-Gaussian models in finance and econometrics. Math. Comput. Modelling 34 (2001), no. 9-11, 1023–1035.
  • Kozubowski, T. J.; Panorska, A. K. Multivariate geometric stable distributions in financial applications. Math. Comput. Modelling 29 (1999), no. 10-12, 83–92.
  • Kozubowski T.J., Rachev S.T.: The theory of geometric stable distributions and its use in modeling financial data, ph% European Journ. Operat. Research., 74, (1994), 310–324.
  • Kuznetsov, Alexey. Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Probab. 20 (2010), no. 5, 1801–1830.
  • Lopez-Mimbela, José Alfredo; Privault, Nicolas. Blow-up and stability of semilinear PDEs with gamma generators. J. Math. Anal. Appl. 307 (2005), no. 1, 181–205.
  • Madan D.B., Carr P.P., Chang E.C.: The Variance Gamma Process and option pricing, phEuropean Finance Review, 2, (1998), 79–105.
  • Mainardi, Francesco; Luchko, Yuri; Pagnini, Gianni. The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4 (2001), no. 2, 153–192.
  • Meerschaert M.M., Benson D.A., Baumer B.: Multidimensional advection and fractional dispersion, phPhys Rev E, 59 (5 A), (1999), 5026–8.
  • Mittnik, Stefan; Rachev, Svetlozar T. Alternative multivariate stable distributions and their applications to financial modeling. Stable processes and related topics (Ithaca, NY, 1990), 107–119, Progr. Probab., 25, Birkhäuser Boston, Boston, MA, 1991.
  • Prabhu, N. U. Wiener-Hopf factorization for convolution semigroups. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), 103–113.
  • Saichev, Alexander I.; Zaslavsky, George M. Fractional kinetic equations: solutions and applications. Chaos 7 (1997), no. 4, 753–764.
  • Samorodnitsky, Gennady; Taqqu, Murad S. Stable non-Gaussian random processes. Stochastic models with infinite variance. Stochastic Modeling. Chapman & Hall, New York, 1994. xxii+632 pp. ISBN: 0-412-05171-0
  • Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4
  • Å ikić, Hrvoje; Song, Renming; Vondraček, Zoran. Potential theory of geometric stable processes. Probab. Theory Related Fields 135 (2006), no. 4, 547–575.