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

Abstract

Let $\{B_t^{(\mu)},t\geqq0\}$ be a one-dimensional Brownian motion with constant drift $\mu\in{\bf R}$ starting from $0$. In this paper we show that $$ Z_t^{(\mu)} = \exp(-B_t^{(\mu)}) \int_0^t \exp(2B_s^{(\mu)}) ds $$ 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 $\{Z_t^{(\mu)},t\geqq0\}$ and, in particular, its relation to the generalized Bessel processes.