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The sum of log-normal variates in geometric Brownian motion

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  • Ole Peters
  • Alexander Adamou

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.

Suggested Citation

  • Ole Peters & Alexander Adamou, 2018. "The sum of log-normal variates in geometric Brownian motion," Papers 1802.02939, arXiv.org.
  • Handle: RePEc:arx:papers:1802.02939
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    File URL: http://arxiv.org/pdf/1802.02939
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    References listed on IDEAS

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    1. Ole Peters & Murray Gell-Mann, 2014. "Evaluating gambles using dynamics," Papers 1405.0585, arXiv.org, revised Jun 2015.
    2. Ole Peters & William Klein, 2012. "Ergodicity breaking in geometric Brownian motion," Papers 1209.4517, arXiv.org, revised Mar 2013.
    3. Jean-Philippe Bouchaud & Marc Mezard, 2000. "Wealth condensation in a simple model of economy," Science & Finance (CFM) working paper archive 500026, Science & Finance, Capital Fund Management.
    4. Jean-Philippe Bouchaud, 2015. "On growth-optimal tax rates and the issue of wealth inequalities," Papers 1508.00275, arXiv.org, revised Aug 2015.
    5. Ole Peters & Alexander Adamou, 2015. "An evolutionary advantage of cooperation," Papers 1506.03414, arXiv.org, revised May 2018.
    6. Bouchaud, Jean-Philippe & Mézard, Marc, 2000. "Wealth condensation in a simple model of economy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 282(3), pages 536-545.
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    Cited by:

    1. Cyprien Grau, 2020. "Stochastic Valuation of Revenue-Collecting Tokens in Cryptoeconomic Organizations," Working Papers hal-02894497, HAL.

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