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

Abstract

Let {B(μ)t,t ≧ 0} be a one-dimensional Brownian motion with constant drift μ ∈ R starting from 0. In this paper we show that Z(μ)t =exp(-B(μ)t) ∫t0 exp(2B(μ)s)ds gives rise to a diffusion process and we explain how this result may be considered as an extension of the celebrated Pitman's 2M - X theorem. We also derive the infinitesimal generator and some properties of the diffusion process {Z(μ)t, t ≧ 0} and, in particular, its relation to the generalized Bessel processes.