Brownian penalisations related to excursion lengths, VII

B. Roynette; P. Vallois; M. Yor

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 2, page 421-452
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Roynette, B., Vallois, P., and Yor, M.. "Brownian penalisations related to excursion lengths, VII." Annales de l'I.H.P. Probabilités et statistiques 45.2 (2009): 421-452. <http://eudml.org/doc/78028>.

@article{Roynette2009,
abstract = {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.},
author = {Roynette, B., Vallois, P., Yor, M.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {longest length of excursions; brownian meander; penalisation; Brownian meander},
language = {eng},
number = {2},
pages = {421-452},
publisher = {Gauthier-Villars},
title = {Brownian penalisations related to excursion lengths, VII},
url = {http://eudml.org/doc/78028},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Roynette, B.
AU - Vallois, P.
AU - Yor, M.
TI - Brownian penalisations related to excursion lengths, VII
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 2
SP - 421
EP - 452
AB - 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.
LA - eng
KW - longest length of excursions; brownian meander; penalisation; Brownian meander
UR - http://eudml.org/doc/78028
ER -

References

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  1. [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. Zbl0743.60045MR1022900
  2. [2] M. Chesney, M. Jeanblanc-Picqué and M. Yor. Brownian excursions and Parisian barrier options. Adv. in Appl. Probab. 29 (1997) 165–184. Zbl0882.60042MR1432935
  3. [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. Zbl1055.60078MR1954864
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  5. [5] Y. Hu and Z. Shi. Extreme lengths in Brownian and Bessel excursions. Bernoulli 3 (1997) 387–402. Zbl0907.60036MR1483694
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  7. [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. Zbl0622.60083MR896740
  8. [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.) Zbl0271.33001
  9. [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. Zbl0323.60039MR205287
  10. [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. Zbl0917.60006MR1725357
  11. [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. Zbl1150.60308MR2162812
  12. [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. Zbl1121.60027MR2229621
  13. [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. Zbl1121.60004MR2253307
  14. [14] B. Roynette, P. Vallois and M. Yor. Some penalisations of the Wiener measure. Japan J. Math. 1 (2006) 263–290. Zbl1160.60315MR2261065
  15. [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. Zbl1164.60355MR2361439
  16. [16] B. Roynette, P. Vallois and M. Yor. Penalizing a Bes(d) process (0&lt;d&lt;2) with a function of its local time at 0, V. Studia Sci. Math. Hungar. 45 (2009), 67–124. Zbl1164.60307MR2401169
  17. [17] B. Roynette, P. Vallois and M. Yor. Penalisations of multidimensional Brownian motion, VI. To appear in ESAIM PS (2009). Zbl1189.60069MR2518544

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