  Geometric Brownian motion is an exemplary stochastic processes obeying
multiplicative noise, with widespread applications in several fields, e.g. in
finance, in physics and biology. The definition of the process depends
crucially on the interpretation of the stochastic integrals which involves the
discretization parameter $\alpha$ with $0 \leq \alpha \leq 1$ , giving rise to
the well-known special cases $\alpha=0$ (It\^{o}), $\alpha=1/2$
(Fisk-Stratonovich) and $\alpha=1$ (H\"{a}nggi-Klimontovich or anti-It\^{o}).
In this paper we study the asymptotic limits of the probability distribution
functions (PDFs) of geometric Brownian motion and some related generalizations.
We establish the conditions for the existence of normalizable asymptotic
distributions depending on the discretization parameter $\alpha$. Using the
infinite ergodicity approach, recently applied to stochastic processes with
multiplicative noise by E. Barkai and collaborators, we show how meaningful
asymptotic results can be formulated in a transparent way.
