Some limiting laws associated with the integrated Brownian motion

TL;DR

This paper investigates limit theorems for integrated Brownian motion with various functionals, demonstrating the universality of the conditioned process obtained via penalization, and providing explicit descriptions of these processes.

Contribution

It establishes the penalization principle for multiple functionals of integrated Brownian motion and shows the invariance of the conditioned process with respect to the passage time index.

Findings

Penalization principle holds for first, nth, last passage times, and supremum.

Conditioned process is integrated Brownian motion conditioned not to hit zero.

Penalization by the nth passage time is independent of n.

Abstract

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 conditioned 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.