A version of Pitman's 2M-X theorem for geometric Brownian motions

We show that geometric Brownian motion with parameter mu, i.e., the exponen tial of linear Brownian motion with drift mu, divided by its quadratic vari ation process is a diffusion process. Taking logarithms and an appropriate scaling limit, we recover the Rogers-Pitman extension to Brownian motion wi th drift of Pitman's representation theorem for the three-dimensional Besse l process. Time inversion and generalized inverse Gaussian distributions pl ay crucial roles in our proofs. (C) Academie des Sciences/Elsevier, Paris.