Brownian penalisations related to excursion lengths, VII
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 2, p. 421-452

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.

DOI : https://doi.org/10.1214/08-AIHP177
Classification:  60F17,  60F99,  60G17,  60G40,  60G44,  60H10,  60H20,  60J25,  60J55,  60J60,  60J65
Keywords: longest length of excursions, brownian meander, penalisation
@article{AIHPB_2009__45_2_421_0,
     author = {Roynette, Bernard and Vallois, Pierre and Yor, Marc},
     title = {Brownian penalisations related to excursion lengths, VII},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {2},
     year = {2009},
     pages = {421-452},
     doi = {10.1214/08-AIHP177},
     zbl = {1181.60046},
     mrnumber = {2521408},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_2_421_0}
}
Roynette, B.; Vallois, P.; Yor, M. Brownian penalisations related to excursion lengths, VII. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 2, pp. 421-452. doi : 10.1214/08-AIHP177. http://www.numdam.org/item/AIHPB_2009__45_2_421_0/

[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. | Numdam | MR 1022900 | Zbl 0743.60045

[2] M. Chesney, M. Jeanblanc-Picqué and M. Yor. Brownian excursions and Parisian barrier options. Adv. in Appl. Probab. 29 (1997) 165-184. | MR 1432935 | Zbl 0882.60042

[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. | MR 1954864 | Zbl 1055.60078

[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. | Numdam | MR 436342 | Zbl 0321.60056

[5] Y. Hu and Z. Shi. Extreme lengths in Brownian and Bessel excursions. Bernoulli 3 (1997) 387-402. | MR 1483694 | Zbl 0907.60036

[6] T. Jeulin. Semi-martingales et grossissement d'une filtration. Lecture Notes in Mathematics 833. Springer, Berlin, 1980. | MR 604176 | Zbl 0444.60002

[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. | MR 896740 | Zbl 0622.60083

[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.) | Zbl 0271.33001

[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. | MR 205287 | Zbl 0323.60039

[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. | MR 1725357 | Zbl 0917.60006

[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. | MR 2162812 | Zbl 1150.60308

[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. | MR 2229621 | Zbl 1121.60027

[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. | MR 2253307 | Zbl 1121.60004

[14] B. Roynette, P. Vallois and M. Yor. Some penalisations of the Wiener measure. Japan J. Math. 1 (2006) 263-290. | MR 2261065 | Zbl 1160.60315

[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. | MR 2361439 | Zbl 1164.60355

[16] B. Roynette, P. Vallois and M. Yor. Penalizing a Bes(d) process (0, V. Studia Sci. Math. Hungar. 45 (2009), 67-124. | MR 2401169 | Zbl 1164.60307

[17] B. Roynette, P. Vallois and M. Yor. Penalisations of multidimensional Brownian motion, VI. To appear in ESAIM PS (2009). | Numdam | MR 2518544 | Zbl pre05660763