A link between Bougerol’s identity and a formula due to Donati-Martin, Matsumoto and Yor

Abstract

We point out an easy link between two striking identities on exponential functionals of the Wiener process and the Wiener bridge originated by Bougerol, and Donati-Martin, Matsumoto and Yor, respectively. The link is established using a continuousone-parameter family of Gaussian processes known as α-Wiener bridges or scaled Wiener bridges, which in case α = 0 coincides with a Wiener process and for α = 1 is a version of the Wiener bridge.