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Large systems of diffusions interacting through their ranks

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  • Shkolnikov, Mykhaylo
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    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.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0304414912000208
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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 122 (2012)
    Issue (Month): 4 ()
    Pages: 1730-1747

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    Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1730-1747

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    Related research

    Keywords: Diffusion processes; McKean–Vlasov equation; Porous medium equation; Particle method; Capital distributions; Rank-based market models;

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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.
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    Cited by:
    1. Shkolnikov, Mykhaylo, 2013. "Large volatility-stabilized markets," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 212-228.

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