IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1608.00814.html
   My bibliography  Save this paper

SPDE limit of the global fluctuations in rank-based models

Author

Listed:
  • Praveen Kolli
  • Mykhaylo Shkolnikov

Abstract

We consider systems of diffusion processes ("particles") interacting through their ranks (also referred to as "rank-based models" in the mathematical finance literature). We show that, as the number of particles becomes large, the process of fluctuations of the empirical cumulative distribution functions converges to the solution of a linear parabolic SPDE with additive noise. The coefficients in the limiting SPDE are determined by the hydrodynamic limit of the particle system which, in turn, can be described by the porous medium PDE. The result opens the door to a thorough investigation of large equity markets and investment therein. In the course of the proof we also derive quantitative propagation of chaos estimates for the particle system.

Suggested Citation

  • Praveen Kolli & Mykhaylo Shkolnikov, 2016. "SPDE limit of the global fluctuations in rank-based models," Papers 1608.00814, arXiv.org.
    • Handle: RePEc:arx:papers:1608.00814
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1608.00814
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Shkolnikov, Mykhaylo, 2012. "Large systems of diffusions interacting through their ranks," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1730-1747.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. David Itkin & Martin Larsson, 2021. "On A Class Of Rank-Based Continuous Semimartingales," Papers 2104.04396, arXiv.org.
    2. 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.
    3. Mykhaylo Shkolnikov & Lane Chun Yeung, 2024. "From rank-based models with common noise to pathwise entropy solutions of SPDEs," Papers 2406.07286, arXiv.org.
    4. 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.
    5. Shkolnikov, Mykhaylo, 2013. "Large volatility-stabilized markets," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 212-228.
    6. Benjamin Jourdain & Julien Reygner, 2015. "Capital distribution and portfolio performance in the mean-field Atlas model," Post-Print hal-00921151, HAL.
    7. Andrey Sarantsev, 2019. "Comparison Techniques for Competing Brownian Particles," Journal of Theoretical Probability, Springer, vol. 32(2), pages 545-585, June.
    8. Marcel Nutz & Yuchong Zhang, 2019. "A Mean Field Competition," Management Science, INFORMS, vol. 44(4), pages 1245-1263, November.
    9. 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.
    10. Alexander Vervuurt, 2015. "Topics in Stochastic Portfolio Theory," Papers 1504.02988, arXiv.org.
    11. 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.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1608.00814. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.