An analogue of Pitman's {$2M-X$} theorem for exponential Wiener functionals. II. The role of the generalized inverse Gaussian laws

Abstract

In Part I of this work, we have shown that the stochastic process $Z^{(\mu)}$ defined by (8.1) below is a diffusion process, which may be considered as an extension of Pitman's $2M-X$ theorem. In this Part II, we deduce from an identity in law partly due to Dufresne that $Z^{(\mu)}$ is intertwined with Brownian motion with drift $\mu$ and that the intertwining kernel may be expressed in terms of Generalized Inverse Gaussian laws.