Infinite ergodicity for geometric Brownian motion

TL;DR

This paper investigates the long-term behavior of geometric Brownian motion under different stochastic integral interpretations, revealing conditions for infinite ergodicity and asymptotic distributions.

Contribution

It establishes the conditions for asymptotic distribution existence in geometric Brownian motion based on the discretization parameter and applies infinite ergodicity theory to clarify long-term behavior.

Findings

Asymptotic distributions depend on the stochastic integral interpretation.

Conditions for normalizable asymptotic distributions are derived.

Infinite ergodicity provides a framework for understanding long-term dynamics.

Abstract

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.