Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On Wiener–Hopf factors for stable processes

Piotr Graczyk and Tomasz Jakubowski

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Abstract

We give a series representation of the logarithm of the bivariate Laplace exponent κ of α-stable processes for almost all α ∈ (0, 2].

Résumé

Nous donnons un développement en série du logarithme de l’exposant de Laplace bivarié κ des processus α-stables pour presque tous α ∈ (0, 2].

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 1 (2011), 9-19.

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1294170227

Digital Object Identifier
doi:10.1214/09-AIHP348

Mathematical Reviews number (MathSciNet)
MR2779394

Zentralblatt MATH identifier
1208.60044

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60E10: Characteristic functions; other transforms

Keywords
Stable process Wiener–Hopf factorization

Citation

Graczyk, Piotr; Jakubowski, Tomasz. On Wiener–Hopf factors for stable processes. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 1, 9--19. doi:10.1214/09-AIHP348. https://projecteuclid.org/euclid.aihp/1294170227


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References

  • [1] A. Baker. A Concise Introduction to the Theory of Numbers. Cambridge Univ. Press, Cambridge, 1984.
  • [2] V. Bernyk, R. C. Dalang and G. Peskir. The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab. 36 (2008) 1777–1789.
  • [3] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996.
  • [4] N. H. Bingham. Maxima of sums of random variables and suprema of stable processes. Z. Wahrsch. Verw. Gebiete 26 (1973) 273–296.
  • [5] F. Caravenna and L. Chaumont. Invariance principles for random walks conditioned to stay positive. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 170–190.
  • [6] L. Chaumont, A. E. Kyprianou and J. C. Pardo. Some explicit identities associated with positive self-similar Markov processes. Stochastic Process. Appl. 119 (2009) 980–1000.
  • [7] D. A. Darling. The maximum of sums of stable random variables. Trans. Amer. Math. Soc. 83 (1956) 164–169.
  • [8] R. A. Doney. On Wiener–Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Probab. 15 (1987) 1352–1362.
  • [9] P. Graczyk and T. Jakubowski. On exit time of symmetric α-stable processes. Preprint, 2009.
  • [10] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products, 7th edition. Elsevier/Academic Press, Amsterdam, 2007.
  • [11] A. Kuznetsov. Wiener–Hopf factorization and distribution of extrema for a family of Lévy processes. J. Appl. Probab. (2009). To appear.
  • [12] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin, 2006.
  • [13] A. E. Kyprianou and Z. Palmowski. Fluctuations of spectrally negative Markov additive processes. In Séminaire de probabilités XLI. Lecture Notes in Math. 1934 121–135. Springer, Berlin, 2008.
  • [14] M. Waldschmidt. Private communication, 2009.