Large systems of diffusions interacting through their ranks
AbstractWe 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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Bibliographic InfoArticle provided by Elsevier in its journal Stochastic Processes and their Applications.
Volume (Year): 122 (2012)
Issue (Month): 4 ()
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Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description
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- Tomoyuki Ichiba & Vassilios Papathanakos & Adrian Banner & Ioannis Karatzas & Robert Fernholz, 2009. "Hybrid Atlas models," Papers 0909.0065, arXiv.org, revised Apr 2011.
- Shkolnikov, Mykhaylo, 2013. "Large volatility-stabilized markets," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 212-228.
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