K-Hartman-Watson distributions: A study on distributional dependencies between functionals of geometric Brownian motion, GIG and Hartman-Watson distributions

Abstract

The new general results for Brownian motion functionals are obtained by introducing the so called K-transforms of the special function u. The transforms lead to the new distributions called here K-Hartman-Watson distributions. They are counterparts to the classical Hartman-Watson distributions, but with respect to the state-space variable. It is shown that the distribution of geometric Brownian motion eBt and its additive functional At are described by the composition of General Inverse Gaussian (GIG) distribution with K-Hartman-Watson distribution. As a consequence two distributional dependencies are presented: the first between the distribution of GIG distribution on R+ and the distribution of functional At, and the second one, between the distribution of GIG on a hyperplane and the distribution of the vector (At,eBt).