The Annals of Probability

On extrema of stable processes

Alexey Kuznetsov

Full-text: Open access


We study the Wiener–Hopf factorization and the distribution of extrema for general stable processes. By connecting the Wiener–Hopf factors with a certain elliptic-like function we are able to obtain many explicit and general results, such as infinite series representations and asymptotic expansions for the density of supremum, explicit expressions for the Wiener–Hopf factors and the Mellin transform of the supremum, quasi-periodicity and functional identities for these functions, finite product representations in some special cases and identities in distribution satisfied by the supremum functional.

Article information

Ann. Probab., Volume 39, Number 3 (2011), 1027-1060.

First available in Project Euclid: 16 March 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G52: Stable processes

Stable processes supremum Wiener–Hopf factorization Mellin transform functional equations elliptic functions double Gamma function q-Pochhammer symbol Clausen function


Kuznetsov, Alexey. On extrema of stable processes. Ann. Probab. 39 (2011), no. 3, 1027--1060. doi:10.1214/10-AOP577.

Export citation


  • [1] Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions. Encyclopedia of Mathematics and Its Applications 71. Cambridge Univ. Press, Cambridge.
  • [2] Barnes, E. W. (1899). The genesis of the double gamma function. Proc. London Math. Soc. 31 358–381.
  • [3] Barnes, E. W. (1901). The theory of the double gamma function. Phil. Trans. Royal Soc. London (A) 196 265–387.
  • [4] Bernyk, V., Dalang, R. C. and Peskir, G. (2008). The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab. 36 1777–1789.
  • [5] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [6] Billingham, J. and King, A. C. (1997). Uniform asymptotic expansions for the Barnes double gamma function. Proc. Roy. Soc. London Ser. A 453 1817–1829.
  • [7] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. in Appl. Probab. 7 705–766.
  • [8] Borodin, A. N. and Salminen, P. (1996). Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel.
  • [9] Borwein, J. M., Bradley, D. M. and Crandall, R. E. (2000). Computational strategies for the Riemann zeta function. J. Comput. Appl. Math. 121 247–296.
  • [10] Darling, D. A. (1956). The maximum of sums of stable random variables. Trans. Amer. Math. Soc. 83 164–169.
  • [11] Doney, R. A. (1987). On Wiener–Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Probab. 15 1352–1362.
  • [12] Doney, R. A. (2008). A note on the supremum of a stable process. Stochastics 80 151–155.
  • [13] Doney, R. A. and Savov, M. S. (2010). The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab. 38 316–326.
  • [14] Fourati, S. (2006). Inversion de l’espace et du temps des processus de Lévy stables. Probab. Theory Related Fields 135 201–215.
  • [15] Graczyk, P. and Jakubowski, T. (2009). Wiener–Hopf factors for stable processes. Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • [16] Heyde, C. C. (1969). On the maximum of sums of random variables and the supremum functional for stable processes. J. Appl. Probab. 6 419–429.
  • [17] Jeffrey, A. ed. (2007). Table of Integrals, Series and Products, 7th ed. Academic Press, Amsterdam.
  • [18] Khinchin, A. Y. (1997). Continued Fractions, Russian ed. Dover Publications Inc., Mineola, NY.
  • [19] Kuznetsov, A. (2009). Analytical proof of Pecherskii–Rogozin identity and Wiener–Hopf factorization. Theory Probab. Appl. To appear.
  • [20] Lawrie, J. B. and King, A. C. (1994). Exact solution to a class of functional difference equations with application to a moving contact line flow. European J. Appl. Math. 5 141–157.
  • [21] Lewin, L. (1981). Polylogarithms and Associated Functions. North-Holland, New York.
  • [22] Lewis, A. L. and Mordecki, E. (2008). Wiener–Hopf factorization for Lévy processes having positive jumps with rational transforms. J. Appl. Probab. 45 118–134.
  • [23] Patie, P. (2009). A few remarks on the supremum of stable processes. Statist. Probab. Lett. 79 1125–1128.
  • [24] Peskir, G. (2008). The law of the hitting times to points by a stable Lévy process with no negative jumps. Electron. Comm. Probab. 13 653–659.
  • [25] Vatutin, V. A. and Wachtel, V. (2009). Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 177–217.
  • [26] Zolotarev, V. M. (1957). Mellin–Stieltjes transformations in probability theory. Teor. Veroyatnost. i Primenen. 2 444–469.

Supplemental materials

  • Supplementary material: Appendix A: Detailed proofs of some results related to the double gamma function. This supplement material provides detailed computations needed to derive formulas (4.10), (4.11), (7.1), (7.2), (7.5) and to prove Corollary 3 and Theorem 8.