  Starting from the hyperbolic Brownian motion as a time-changed Brownian
motion, we explore a set of probabilistic models--related to the SABR model in
mathematical finance--which can be obtained by geometry-preserving
transformations, and show how to translate the properties of the hyperbolic
Brownian motion (density, probability mass, drift) to each particular model.
Our main result is an explicit expression for the probability of any of these
models hitting the boundary of their domains, the proof of which relies on the
properties of the aforementioned transformations as well as time-change
methods.
