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

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