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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 continuous one-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.

A Link Between Bougerol’s Identity and a Formula Due to Donati-Martin, Matsumoto and Yor