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Étude des trajectoires de la primitive du mouvement brownien

Abstract : In this work we gather several results we obtained on the behavior of the integral of linear Brownian motion, and more particularly on the various distributions related to the first passage times of the trajectories by fixed thresholds. For instance, we were able to explicitly determine the joint law of the couple made up of the first passage time of the integrated process by a fixed point and of the related location of Brownian motion. We retrieved in particular the marginal laws of this couple discovered by M. Goldman (1971) and Ju. P. Gor'kov (1975), as well as the law of the first return time to the origin obtained by H.P. McKean (1963). This result enabled us to resolve several open problems. In particular, we obtained the distributions of several functionals associated with the integral of Brownian motion: successive passage times, last passage time, sojourn time, excursions... We next studied the location of the primitive of Brownian motion when this latter reaches a single or double barrier. Such functionals naturally arise in some optimization problems studied by M. Lefèbvre (1989). A new approach enabled us to find and improve its results. We finally derived the distribution of certain functionals related to the integral of Brownian motion, this latter being subjected to a parabolic or cubic drift. We retrieved in particular a result of P. Groeneboom (1989) concerning Brownian motion with a parabolic drift. An exhibition of some still open problems completes this work.
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Contributor : Aimé Lachal <>
Submitted on : Thursday, August 27, 2020 - 7:30:16 PM
Last modification on : Friday, September 4, 2020 - 9:45:22 AM


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  • HAL Id : tel-02924274, version 1


Aimé Lachal. Étude des trajectoires de la primitive du mouvement brownien. Probabilités [math.PR]. Université Lyon 1 - Claude Bernard, 1992. Français. ⟨tel-02924274⟩



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