  The discrete sum of geometric Brownian motions plays an important role in
modeling stochastic annuities in insurance. It also plays a pivotal role in the
pricing of Asian options in mathematical finance. In this paper, we study the
probability distributions of the infinite sum of geometric Brownian motions,
the sum of geometric Brownian motions with geometric stopping time, and the
finite sum of the geometric Brownian motions. These results are extended to the
discrete sum of the exponential L\'evy process. We derive tail asymptotics and
compute numerically the asymptotic distribution function. We compare the
results against the known results for the continuous time integral of the
geometric Brownian motion up to an exponentially distributed time. The results
are illustrated with numerical examples for life annuities with discrete
payments, and Asian options.
