On some exponential integral functionals of BM($\mu$) and BES(3)

Abstract

In this paper we derive the Laplace transforms of the integral functionals $$ \int_0^\infty (p(\exp(B^{(\mu)}_t)+1)^{-1}+ q(\exp(B^{(\mu)}_t)+1)^{-2}) dt, $$ $$ \int_0^\infty (p(\exp(R^{(3)}_t)-1)^{-1}+ q(\exp(R^{(3)}_t)-1)^{-2}) dt, $$ where $p$ and $q$ are real numbers, $\{B^{(\mu)}_t: t\geq 0\}$ is a Brownian motion with drift $\mu>0,$ BM($\mu$), and $\{R^{(3)}_t: t\geq 0\}$ is a 3-dimensional Bessel process, BES(3). The transforms are given in terms of Gauss' hypergeometric functions and it is seen that the results are closely related to some functionals of Jacobi diffusions. This work generalizes and completes some results of Donati--Martin and Yor and Salminen and Yor.