Moment generating function of the reciprocal of an integral of geometric Brownian motion

In this paper we obtain a simple, explicit integral form for the moment generating function of the reciprocal of the random variable defined by $A^{(\nu)}_t := \int ^t _0 \exp (2B_s + 2 \nu s) ds$, where $B_s$, $s>0$, is a one-dimensional Brownian motion starting from 0. In case $\nu = 1$, the moment generating function has a particularly simple form.