Volume 19, 2015
|Page(s)||148 - 171|
|Published online||19 August 2015|
Some limiting laws associated with the integrated Brownian motion∗
Laboratoire de Mathématiques et Modélisation d’Evry (LaMME), Université
d’Evry-Val-d’Essonne, UMR CNRS 8071, 91037 Evry cedex, France.
Revised: 11 December 2013
We study some limit theorems for the normalized law of integrated Brownian motion perturbed by several examples of functionals: the first passage time, the nth passage time, the last passage time up to a finite horizon and the supremum. We show that the penalization principle holds in all these cases and give descriptions of the conditioned processes. In particular, it is remarkable that the penalization by the nth passage time is independent of n, and always gives the same penalized process, i.e. integrated Brownian motion conditioned not to hit 0. Our results rely on some explicit formulae obtained by Lachal and on enlargement of filtrations.
Mathematics Subject Classification: 60J65 / 60G15 / 60G44
Key words: Integrated Brownian motion / penalization / passage times
© EDP Sciences, SMAI 2015
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