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Geometric Brownian Motion under Stochastic Resetting: A Stationary yet Non-ergodic Process

Author

Listed:
  • Viktor Stojkoski
  • Trifce Sandev
  • Ljupco Kocarev
  • Arnab Pal

Abstract

We study the effects of stochastic resetting on geometric Brownian motion (GBM), a canonical stochastic multiplicative process for non-stationary and non-ergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains non-ergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the non-ergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.

Suggested Citation

  • Viktor Stojkoski & Trifce Sandev & Ljupco Kocarev & Arnab Pal, 2021. "Geometric Brownian Motion under Stochastic Resetting: A Stationary yet Non-ergodic Process," Papers 2104.01571, arXiv.org, revised Aug 2021.
  • Handle: RePEc:arx:papers:2104.01571
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    File URL: http://arxiv.org/pdf/2104.01571
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    Citations

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    Cited by:

    1. Xuewen Tan & Pengpeng Liu & Wenhui Luo & Hui Chen, 2022. "Analysis of a Class of Predation-Predation Model Dynamics with Random Perturbations," Mathematics, MDPI, vol. 10(18), pages 1-12, September.
    2. Petar Jolakoski & Arnab Pal & Trifce Sandev & Ljupco Kocarev & Ralf Metzler & Viktor Stojkoski, 2022. "The fate of the American dream: A first passage under resetting approach to income dynamics," Papers 2212.13176, arXiv.org.
    3. Viktor Stojkoski & Sonja Mitikj & Marija Trpkova-Nestorovska & Dragan Tevdovski, 2023. "Income Mobility and Mixing in North Macedonia," Papers 2309.17268, arXiv.org, revised Oct 2023.
    4. Viktor Stojkoski & Petar Jolakoski & Arnab Pal & Trifce Sandev & Ljupco Kocarev & Ralf Metzler, 2021. "Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity," Papers 2109.01822, arXiv.org.
    5. Guo, Wei & Liu, Ying-Zhou & Huang, Fei-Jie & Shi, Hong-Da & Du, Lu-Chun, 2023. "Brownian particles in a periodic potential corrugated by disorder: Anomalous diffusion and ergodicity breaking," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    6. Stojkoski, Viktor, 2024. "Measures of physical mixing evaluate the economic mobility of the typical individual," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    7. Jolakoski, Petar & Pal, Arnab & Sandev, Trifce & Kocarev, Ljupco & Metzler, Ralf & Stojkoski, Viktor, 2023. "A first passage under resetting approach to income dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    8. Rajeev Rajaram & Nathan Ritchey, 2023. "Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy," Mathematics, MDPI, vol. 11(15), pages 1-16, July.

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