Revisiting integral functionals of geometric Brownian motion

Abstract

In this paper we revisit the integral functional of geometric Brownian motion $I_t={\int }_0^t{e}^{-(\mu s+\sigma W_s)}ds$, where $\mu \in \mathbb{R}$, $\sigma >0$, and $(W_s{)}_s>0$ is a standard Brownian motion. Specifically, we calculate the Laplace transform in $t$ of the cumulative distribution function and of the probability density function of this functional.