On the probability of hitting the boundary for brownian motions on the SABR plane

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.