Infinite ergodicity for geometric Brownian motion

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.