The sum of log-normal variates in geometric Brownian motion

TL;DR

This paper investigates the distribution of sums of log-normal variates in geometric Brownian motion, revealing new analytical methods and connections to spin glass models to understand their typical trajectories.

Contribution

It introduces two methods—one based on spin glass theory and another using Ito calculus—to analyze sums of log-normal variates in GBM, providing new insights and quantitative results.

Findings

The free energy approach qualitatively predicts GBM sum behavior.

Ito calculus yields results in close agreement with numerical simulations.

The connection to spin glasses offers a novel perspective on GBM sums.

Abstract

Geometric Brownian motion (GBM) is a key model for representing self-reproducing entities. Self-reproduction may be considered the definition of life [5], and the dynamics it induces are of interest to those concerned with living systems from biology to economics. Trajectories of GBM are distributed according to the well-known log-normal density, broadening with time. However, in many applications, what's of interest is not a single trajectory but the sum, or average, of several trajectories. The distribution of these objects is more complicated. Here we show two different ways of finding their typical trajectories. We make use of an intriguing connection to spin glasses: the expected free energy of the random energy model is an average of log-normal variates. We make the mapping to GBM explicit and find that the free energy result gives qualitatively correct behavior for GBM trajectories. We then also compute the typical sum of lognormal variates using Ito calculus. This alternative route is in close quantitative agreement with numerical work.