Ergodicity breaking in geometric Brownian motion

Abstract

Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time-average. A common tactic for bringing time averages closer to ensemble averages is diversification. In this letter we study the effects of diversification using the concept of ergodicity breaking.

5 pages, 3 figures

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• Colloquium: Statistical mechanics of money, wealth, and income
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• Optimal leverage from non-ergodicity
• The time resolution of the St. Petersburg paradox

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