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.
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.
Mots clés : longest length of excursions, brownian meander, penalisation
@article{AIHPB_2009__45_2_421_0, author = {Roynette, B. and Vallois, P. and Yor, M.}, title = {Brownian penalisations related to excursion lengths, {VII}}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {421--452}, publisher = {Gauthier-Villars}, volume = {45}, number = {2}, year = {2009}, doi = {10.1214/08-AIHP177}, mrnumber = {2521408}, zbl = {1181.60046}, language = {en}, url = {http://www.numdam.org/articles/10.1214/08-AIHP177/} }
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 SP - 421 EP - 452 VL - 45 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/08-AIHP177/ DO - 10.1214/08-AIHP177 LA - en ID - AIHPB_2009__45_2_421_0 ER -
%0 Journal Article %A Roynette, B. %A Vallois, P. %A Yor, M. %T Brownian penalisations related to excursion lengths, VII %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 421-452 %V 45 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/08-AIHP177/ %R 10.1214/08-AIHP177 %G en %F 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, Tome 45 (2009) no. 2, pp. 421-452. doi : 10.1214/08-AIHP177. http://www.numdam.org/articles/10.1214/08-AIHP177/
[1] Étude d'une martingale remarquable. In Séminaire de Probabilités, XXIII 88-130. Lecture Notes in Math. 1372. Springer, Berlin, 1989. | Numdam | MR | Zbl
and .[2] Brownian excursions and Parisian barrier options. Adv. in Appl. Probab. 29 (1997) 165-184. | MR | Zbl
, and .[3] On independent times and positions for Brownian motions. Rev. Mat. Iberoamericana 18 (2002) 541-586. | MR | Zbl
, , and .[4] 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 | Zbl
.[5] Extreme lengths in Brownian and Bessel excursions. Bernoulli 3 (1997) 387-402. | MR | Zbl
and .[6] Semi-martingales et grossissement d'une filtration. Lecture Notes in Mathematics 833. Springer, Berlin, 1980. | MR | Zbl
.[7] 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 | Zbl
.[8] 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
.[9] 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 | Zbl
.[10] Continuous Martingales and Brownian Motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin, 1999. | MR | Zbl
and .[11] Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III. Period. Math. Hungar. 50 (2005) 247-280. | MR | Zbl
, and .[12] Limiting laws associated with Brownian motion perturbed by normalized exponential weights, I. Studia Sci. Math. Hungar. 43 (2006) 171-246. | MR | Zbl
, and .[13] Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II. Studia Sci. Math. Hungar. 43 (2006) 295-360. | MR | Zbl
, and .[14] Some penalisations of the Wiener measure. Japan J. Math. 1 (2006) 263-290. | MR | Zbl
, and .[15] 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 | Zbl
, and .[16] 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. | MR | Zbl
, and .[17] Penalisations of multidimensional Brownian motion, VI. To appear in ESAIM PS (2009). | Numdam | MR
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