  The aim of this paper is to investigate discrete approximations of the
exponential functional $\int_0^{\infty} \exp(B(t) - \nu t) \di t$ of Brownian
motion (which plays an important role in Asian options of financial
mathematics) by the help of simple, symmetric random walks. In some
applications the discrete model could be even more natural than the continuous
one. The properties of the discrete exponential functional are rather different
from the continuous one: typically its distribution is singular w.r.t. Lebesgue
measure, all of its positive integer moments are finite and they characterize
the distribution. On the other hand, using suitable random walk approximations
to Brownian motion, the resulting discrete exponential functionals converge
a.s. to the exponential functional of Brownian motion, hence their limit
distribution is the same as in the continuous case, namely, the one of the
reciprocal of a gamma random variable, so absolutely continuous w.r.t. Lebesgue
measure. This way we give a new, elementary proof for an earlier result by
Dufresne and Yor as well.
