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Large systems of diffusions interacting through their ranks

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  • Shkolnikov, Mykhaylo

Abstract

We study the limiting behavior of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that under certain assumptions the limiting dynamics is given by a McKean–Vlasov evolution equation. Moreover, we show that the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The implications of the results for rank-based models of capital distributions in financial markets are also explained.

Suggested Citation

  • Shkolnikov, Mykhaylo, 2012. "Large systems of diffusions interacting through their ranks," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1730-1747.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1730-1747
    DOI: 10.1016/j.spa.2012.01.011
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    References listed on IDEAS

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    1. Tomoyuki Ichiba & Vassilios Papathanakos & Adrian Banner & Ioannis Karatzas & Robert Fernholz, 2009. "Hybrid Atlas models," Papers 0909.0065, arXiv.org, revised Apr 2011.
    2. B. Jourdain, 2000. "Diffusion Processes Associated with Nonlinear Evolution Equations for Signed Measures," Methodology and Computing in Applied Probability, Springer, vol. 2(1), pages 69-91, April.
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    Cited by:

    1. Andrey Sarantsev, 2017. "Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1200-1223, September.
    2. Marcel Nutz & Yuchong Zhang, 2019. "A Mean Field Competition," Management Science, INFORMS, vol. 44(4), pages 1245-1263, November.
    3. David Itkin & Martin Larsson, 2021. "On A Class Of Rank-Based Continuous Semimartingales," Papers 2104.04396, arXiv.org.
    4. Benjamin Jourdain & Julien Reygner, 2015. "Capital distribution and portfolio performance in the mean-field Atlas model," Annals of Finance, Springer, vol. 11(2), pages 151-198, May.
    5. Andrey Sarantsev, 2019. "Comparison Techniques for Competing Brownian Particles," Journal of Theoretical Probability, Springer, vol. 32(2), pages 545-585, June.
    6. Sergio A. Almada Monter & Mykhaylo Shkolnikov & Jiacheng Zhang, 2018. "Dynamics of observables in rank-based models and performance of functionally generated portfolios," Papers 1802.03593, arXiv.org.
    7. Shkolnikov, Mykhaylo, 2013. "Large volatility-stabilized markets," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 212-228.
    8. Benjamin Jourdain & Julien Reygner, 2015. "Capital distribution and portfolio performance in the mean-field Atlas model," Post-Print hal-00921151, HAL.
    9. Alexander Vervuurt, 2015. "Topics in Stochastic Portfolio Theory," Papers 1504.02988, arXiv.org.
    10. Praveen Kolli & Mykhaylo Shkolnikov, 2016. "SPDE limit of the global fluctuations in rank-based models," Papers 1608.00814, arXiv.org.
    11. Mykhaylo Shkolnikov & Lane Chun Yeung, 2024. "From rank-based models with common noise to pathwise entropy solutions of SPDEs," Papers 2406.07286, arXiv.org.
    12. Brandon Flores & Blessing Ofori-Atta & Andrey Sarantsev, 2021. "A stock market model based on CAPM and market size," Annals of Finance, Springer, vol. 17(3), pages 405-424, September.

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