An analogue of Pitman's 2M-X theorem for exponential Wiener functionals Part I: A time-inversion approach

Let {B-t((mu)),t greater than or equal to 0} be a one-dimensional Brownian motion with constant drift mu is an element of R starting from 0. In this p aper we show that Z(t)((mu)) = exp(-B-t((mu))) integral(0)(t) exp(2B(s)((mu)))ds gives rise to a diffusion process and we explain how this result may be con sidered as an extension of the celebrated Pitman's 2M - X theorem. We also derive the infinitesimal generator and some properties of the diffusion pro cess {Z(t)((mu)), t greater than or equal to 0} and, in particular, its rel ation to the generalized Bessel processes.