Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the asymptotic behavior of the density of the supremum of Lévy processes

Loïc Chaumont and Jacek Małecki

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let us consider a real valued Lévy process $X$ whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of $X$ before any deterministic time $t$ is absolutely continuous on $(0,\infty)$. We show that its density $f_{t}(x)$ is continuous on $(0,\infty)$ if and only if the potential density $h'$ of the upward ladder height process is continuous on $(0,\infty)$. Then we prove that $f_{t}$ behaves at 0 as $h'$. We also describe the asymptotic behaviour of $f_{t}$, when $t$ tends to infinity. Then some related results are obtained for the density of the meander and this of the entrance law of the Lévy process conditioned to stay positive.


Soit $X$ un processus de Lévy réel dont les probabilités de transition sont absolument continues par rapport à la mesure de Lebesgue. Dans ce cas, il est connu que la loi du supremum passé avant un temps déterministe $t$ est elle-même absolument continue sur $(0,\infty)$. En supposant de plus que les densités sont bornées, nous montrons que la densité $f_{t}(x)$ du supremum passé est continue en $x$ sur $(0,\infty)$, si et seulement si la densité potentielle $h'(x)$ du subordinateur des hauteurs d’échelle ascendant est continue sur $(0,\infty)$. Nous montrons alors que $f_{t}$ se comporte en 0 de la même manière que $h'$. Nous donnons également une description du comportement asymptotique de $f_{t}$, lorsque $t$ tend vers l’infini. Enfin nous appliquons ces résultats pour étudier le comportement asymptotique de la densité du méandre des processus de Lévy et de la densité de la loi d’entrée des processus de Lévy conditionnés à rester positifs.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1178-1195.

Received: 9 November 2013
Revised: 26 February 2015
Accepted: 2 March 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60J75: Jump processes

Density Past supremum Asymptotic behaviour Renewal function Conditioning to stay positive Meander


Chaumont, Loïc; Małecki, Jacek. On the asymptotic behavior of the density of the supremum of Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1178--1195. doi:10.1214/15-AIHP674.

Export citation


  • [1] J. Bertoin. Lévy Processes. Cambridge Univ. Press, Cambridge, 1996.
  • [2] L. Chaumont. On the law of the supremum of Lévy processes. Ann. Probab. 41 (3A) (2013) 1191–1217.
  • [3] L. Chaumont and R. A. Doney. On Lévy processes conditionned to stay positive. Electron. J. Probab. 10 (28) (2005) 948–961.
  • [4] L. Chaumont and R. A. Doney. Corrections to “On Lévy processes conditionned to stay positive.” Electron. J. Probab. 13 (28) (2008) 1–4.
  • [5] L. Chaumont and R. A. Doney. Invariance principles for local times at the maximum of random walks and Lévy processes. Ann. Probab. 38 (4) (2010) 1368–1389.
  • [6] L. Chaumont and G. Uribe Bravo. Markovian bridges: Weak continuity and pathwise constructions. Ann. Probab. 39 (2) (2011) 609–647.
  • [7] R. A. Doney. Fluctuation Theory for Lévy Processes. Lectures from the 35th Summer School on Probability Theory Held in Saint-Flour, July 6–23, 2005. Lecture Notes in Math. 1897. Springer, Berlin, 2007. With a foreword by Jean Picard.
  • [8] R. A. Doney and V. Rivero. Asymptotic behaviour of first passage time distributions for Lévy processes. Probab. Theory Related Fields 157 (2013) 1–45.
  • [9] R. A. Doney and M. S. Savov. The asymptotic behavior of densities related to the supremum of a stable process. Ann. Probab. 38 (1) (2010) 316–326.
  • [10] R. K. Getoor. Excursions of a Markov process. Ann. Probab. 7 (2) (1979) 244–266.
  • [11] P. E. Greenwood and A. A. Novikov. One-sided boundary crossing for processes with independent increments. Teor. Veroyatn. Primen. 31 (2) (1986) 266–277.
  • [12] F. B. Knight. The uniform law for exchangeable and Lévy process bridges. Hommage à P. A. Meyer et J. Neveu. Astérisque 236 (1996) 171–188.
  • [13] A. Kuznetsov, A. E. Kyprianou and V. Rivero. The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II 97–186. Lecture Notes in Math. 2061. Springer, Heidelberg, 2012.
  • [14] A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin, 2006.
  • [15] M. Kwaśnicki, J. Małecki and M. Ryznar. Suprema of Lévy processes. Ann. Probab. 41 (3B) (2013) 2047–2065.
  • [16] P. Lévy. Processus Stochastiques et Mouvement Brownien. Reprint of the second (1965) edition. Les Grands Classiques Gauthier-Villars, Paris. Éditions Jacques Gabay. Sceaux, 1992.
  • [17] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge, 2013.
  • [18] R. Schilling, R. Song and Z. Vondraček. Bernstein Functions: Theory and Applications. Studies in Math. 37. De Gruyter, Berlin, 2010.
  • [19] M. Sharpe. Supports of convolution semigroups and densities. In Probability Measures on Groups and Related Structures XI (Oberwolfach, 1994) 364–369. World Sci. Publ., River Edge, NJ, 1995.
  • [20] M. L. Silverstein. Classification of coharmonic and coinvariant functions for a Lévy process. Ann. Probab. 8 (1980) 539–575.
  • [21] G. Uribe Bravo. Bridges of Lévy processes conditioned to stay positive. Bernoulli 20 (1) (2014) 190–206.