Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Brownian penalisations related to excursion lengths, VII

B. Roynette, P. Vallois, and M. Yor

Full-text: Open access


Limiting laws, as t→∞, for Brownian motion penalised by the longest length of excursions up to t, or up to the last zero before t, or again, up to the first zero after t, are shown to exist, and are characterized.


Il est prouvé que les lois limites, lorsque t→∞, du mouvement brownien pénalisé par la plus grande longueur des excursions jusqu’en t, ou bien jusqu’au dernier zéro avant t, ou encore jusqu’au premier zéro après t, existent. Ces lois limites sont décrites en détail.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 2 (2009), 421-452.

First available in Project Euclid: 29 April 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60F99: None of the above, but in this section 60G17: Sample path properties 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G44: Martingales with continuous parameter 60H10: Stochastic ordinary differential equations [See also 34F05] 60H20: Stochastic integral equations 60J25: Continuous-time Markov processes on general state spaces 60J55: Local time and additive functionals 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]

Longest length of excursions Brownian meander Penalisation


Roynette, B.; Vallois, P.; Yor, M. Brownian penalisations related to excursion lengths, VII. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 2, 421--452. doi:10.1214/08-AIHP177.

Export citation


  • [1] J. Azéma and M. Yor. Étude d’une martingale remarquable. In Séminaire de Probabilités, XXIII 88–130. Lecture Notes in Math. 1372. Springer, Berlin, 1989.
  • [2] M. Chesney, M. Jeanblanc-Picqué and M. Yor. Brownian excursions and Parisian barrier options. Adv. in Appl. Probab. 29 (1997) 165–184.
  • [3] B. deMeyer, B. Roynette, P. Vallois and M. Yor. On independent times and positions for Brownian motions. Rev. Mat. Iberoamericana 18 (2002) 541–586.
  • [4] R. K. Getoor. On the construction of kernels. In Séminaire de Probabilités, IX (Seconde Partie, Univ. Strasbourg, Strasbourg, années universitaires 1973/1974 et 1974/1975) 443–463. Lecture Notes in Math. 465. Springer, Berlin, 1975.
  • [5] Y. Hu and Z. Shi. Extreme lengths in Brownian and Bessel excursions. Bernoulli 3 (1997) 387–402.
  • [6] T. Jeulin. Semi-martingales et grossissement d’une filtration. Lecture Notes in Mathematics 833. Springer, Berlin, 1980.
  • [7] F. B. Knight. On the duration of the longest excursion. In Seminar on Stochastic Processes, 1985 (Gainesville, Fla., 1985) 117–147. Progr. Probab. Statist. 12. Birkhäuser Boston, Boston, MA, 1986.
  • [8] N. N. Lebedev. Special Functions and Their Applications. Dover, New York, 1972. (Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication.)
  • [9] P. A. Meyer. Probabilités et potentiel. In Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIV. Actualités Scientifiques et Industrielles 1318. Hermann, Paris, 1966.
  • [10] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999.
  • [11] B. Roynette, P. Vallois and M. Yor. Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III. Period. Math. Hungar. 50 (2005) 247–280.
  • [12] B. Roynette, P. Vallois and M. Yor. Limiting laws associated with Brownian motion perturbed by normalized exponential weights, I. Studia Sci. Math. Hungar. 43 (2006) 171–246.
  • [13] B. Roynette, P. Vallois and M. Yor. Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II. Studia Sci. Math. Hungar. 43 (2006) 295–360.
  • [14] B. Roynette, P. Vallois and M. Yor. Some penalisations of the Wiener measure. Japan J. Math. 1 (2006) 263–290.
  • [15] B. Roynette, P. Vallois and M. Yor. Some extensions of Pitman’s and Ray–Knight’s theorems for penalized Brownian motions and their local times, IV. Studia Sci. Math. Hungar. 44 (2007) 469–516.
  • [16] B. Roynette, P. Vallois and M. Yor. Penalizing a Bes(d) process (0<d<2) with a function of its local time at 0, V. Studia Sci. Math. Hungar. 45 (2009), 67–124.
  • [17] B. Roynette, P. Vallois and M. Yor. Penalisations of multidimensional Brownian motion, VI. To appear in ESAIM PS (2009).