Path transformations of first passage bridges. (English) Zbl 1061.60083

Bertoin, Jean; Chaumont, Loïc; Pitman, Jim

Electron. Commun. Probab. 8, 155-166 (2003).

Summary: We define the first passage bridge from 0 to \(\lambda\) as the Brownian motion on the time interval \([0,1]\) conditioned to first hit \(\lambda\) at time 1. We show that this process may be related to the Brownian bridge, the Bessel bridge or the Brownian excursion via some path transformations, the main one being an extension of Vervaat’s transformation. We also propose an extension of these results to certain bridges with cyclically exchangeable increments.

MSC:

60J65	Brownian motion
60G09	Exchangeability for stochastic processes

Keywords:

Brownian bridge; Bessel bridge; Brownian excursion; path transformations