Boundary Crossing Identities for Diffusions Having the Time Inversion Property

We review and study a one-parameter functional transformation, denoted by (S)β∈R, which allow, in the case β < 0, to provide a path realization of bridges of the family of diffusion processes which enjoy the time inversion property. This family includes the Brownian motion, the family of Bessel processes with a positive dimension and their conservative h-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary crossing times for these diffusions over a given function f to those over the image of f by the mapping S, for some fixed β ∈ R. We give some new examples of the boundary crossing problems for the Brownian motion and the family of Bessel processes. In the case of a Brownian motion, we also provide an interpretation of the results obtained by the standard method of images and establish connections between the exact asymptotics for large time of the densities corresponding to various curves of each family.