Comptes Rendus
no. 3
Probability Theory
On the first crossing times of a Brownian motion and a family of continuous curves
[Sur les premiers instants de croisement du mouvement brownien et d'une famille de courbes continues]

Présenté par : Marc Yor

Larbi Alili1 ; Pierre Patie2
1 Department of Statistics, The University of Warwick, Coventry CV4 7AL, UK
2 RiskLab, Department of Mathematics, ETH-Zentrum, CH-8092 Zürich, Switzerland
Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 225-228.

Nous examinons les transformations analytiques qui permettent de passer du mouvement brownien aux ponts browniens standards. Nous les généralisons et étudions certaines de leurs propriétés. L'image d'une courbe réelle et continue, par ces transformations, est une famille de courbes à laquelle nous nous intéressons. Nous établissons une relation simple et explicite entre les distributions des temps d'atteinte de chacun des éléments de cette famille par un mouvement brownien.

We review the analytic transformations allowing to construct standard Brownian bridges from a Brownian motion. These are generalized and some of their properties are studied. The new family maps the space of continuous positive functions into a family of curves which is the topic of our study. We establish a simple and explicit formula relating the distributions of the first hitting times of each of these curves by a standard Brownian motion.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.11.008
Larbi Alili 1 ; Pierre Patie 2

1 Department of Statistics, The University of Warwick, Coventry CV4 7AL, UK
2 RiskLab, Department of Mathematics, ETH-Zentrum, CH-8092 Zürich, Switzerland 
@article{CRMATH_2005__340_3_225_0,
     author = {Larbi Alili and Pierre Patie},
     title = {On the first crossing times of a {Brownian} motion and a family of continuous curves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {225--228},
     publisher = {Elsevier},
     volume = {340},
     number = {3},
     year = {2005},
     doi = {10.1016/j.crma.2004.11.008},
     language = {en},
}
TY  - JOUR
AU  - Larbi Alili
AU  - Pierre Patie
TI  - On the first crossing times of a Brownian motion and a family of continuous curves
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 225
EP  - 228
VL  - 340
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2004.11.008
LA  - en
ID  - CRMATH_2005__340_3_225_0
ER  - 
%0 Journal Article
%A Larbi Alili
%A Pierre Patie
%T On the first crossing times of a Brownian motion and a family of continuous curves
%J Comptes Rendus. Mathématique
%D 2005
%P 225-228
%V 340
%N 3
%I Elsevier
%R 10.1016/j.crma.2004.11.008
%G en
%F CRMATH_2005__340_3_225_0
Larbi Alili; Pierre Patie. On the first crossing times of a Brownian motion and a family of continuous curves. Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 225-228. doi : 10.1016/j.crma.2004.11.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.11.008/

[1] L. Breiman First exit times from a square root boundary, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, CA, 1965/66), vol. II: Contributions to Probability Theory, Part 2, 1967, pp. 9-16

[2] J. Durbin The first-passage density of a continuous Gaussian process to a general boundary, J. Appl. Probab., Volume 22 (1985), pp. 99-122

[3] P. Groeneboom Brownian motion with a parabolic drift and Airy functions, Probab. Theory Related Fields, Volume 81 (1989) no. 1, pp. 79-109

[4] J. Keilson; H.F. Ross Passage time distributions for Gaussian Markov (Ornstein–Uhlenbeck) statistical processes, Selected Tables in Mathematical Studies, III, Amer. Math. Soc., Providence, RI, 1975, pp. 233-327

[5] H.R. Lerche Boundary Crossing of Brownian Motion. Its Relation to the Law of the Iterated Logarithm and to Sequential Analysis, Lecture Notes in Statist., vol. 40, Springer-Verlag, Berlin, 1986

[6] A.A. Novikov, On estimates and the asymptotic bahaviour of non-exit probabilities of a Wiener process to a moving boundary, 1981, pp. 495–505

[7] G. Peskir Limit at zero of the Brownian first-passage density, Probab. Theory Rel. Fields, Volume 124 (2002) no. 1, pp. 100-111

[8] K. Pötzelberger; L. Wang Boundary crossing probability for Brownian motion, J. Appl. Probab., Volume 38 (2001), pp. 152-164

[9] D. Revuz; M. Yor Continuous Martingales and Brownian Motion, vol. 293, Springer-Verlag, Berlin, 1999

[10] P. Salminen On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary, Adv. Appl. Probab., Volume 20 (1988) no. 2, pp. 411-426

[11] V. Strassen Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, CA, 1965/66), vol. II, 1967

Cité par Sources :

Commentaires - Politique