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Ergodicity breaking in geometric Brownian motion

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  • Ole Peters
  • William Klein

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.

Suggested Citation

  • Ole Peters & William Klein, 2012. "Ergodicity breaking in geometric Brownian motion," Papers 1209.4517, arXiv.org, revised Mar 2013.
    • Handle: RePEc:arx:papers:1209.4517
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    Citations

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    Cited by:

    1. Ivan S. Maksymov, 2023. "Analogue and Physical Reservoir Computing Using Water Waves: Applications in Power Engineering and Beyond," Energies, MDPI, vol. 16(14), pages 1-26, July.
    2. Máté, Gabriell & Néda, Zoltán, 2016. "The advantage of inhomogeneity — Lessons from a noise driven linearized dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 445(C), pages 310-317.
    3. Ole Peters & Alexander Adamou, 2018. "The sum of log-normal variates in geometric Brownian motion," Papers 1802.02939, arXiv.org.
    4. Simon Gluzman, 2023. "Market Crashes and Time-Translation Invariance," FinTech, MDPI, vol. 2(2), pages 1-27, March.
    5. De Domenico, Federica & Livan, Giacomo & Montagna, Guido & Nicrosini, Oreste, 2023. "Modeling and simulation of financial returns under non-Gaussian distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).
    6. Ole Peters & Murray Gell-Mann, 2014. "Evaluating gambles using dynamics," Papers 1405.0585, arXiv.org, revised Jun 2015.
    7. Viktor Stojkoski & Trifce Sandev & Lasko Basnarkov & Ljupco Kocarev & Ralf Metzler, 2020. "Generalised geometric Brownian motion: Theory and applications to option pricing," Papers 2011.00312, arXiv.org.
    8. Carlos Rodríguez Raposo & Pablo Coello Pulido, 2021. "Ergodicity transformation for additive-ruin wealth dynamic," Working Papers hal-03198073, HAL.
    9. Federica De Domenico & Giacomo Livan & Guido Montagna & Oreste Nicrosini, 2023. "Modeling and Simulation of Financial Returns under Non-Gaussian Distributions," Papers 2302.02769, arXiv.org.
    10. Dyer, Joel & Cannon, Patrick & Farmer, J. Doyne & Schmon, Sebastian M., 2024. "Black-box Bayesian inference for agent-based models," Journal of Economic Dynamics and Control, Elsevier, vol. 161(C).
    11. Ole Peters & Alexander Adamou, 2015. "An evolutionary advantage of cooperation," Papers 1506.03414, arXiv.org, revised May 2018.

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