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From rank-based models with common noise to pathwise entropy solutions of SPDEs

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  • Mykhaylo Shkolnikov
  • Lane Chun Yeung

Abstract

We study the mean field limit of a rank-based model with common noise, which arises as an extension to models for the market capitalization of firms in stochastic portfolio theory. We show that, under certain conditions on the drift and diffusion coefficients, the empirical cumulative distribution function converges to the solution of a stochastic PDE. A key step in the proof, which is of independent interest, is to show that any solution to an associated martingale problem is also a pathwise entropy solution to the stochastic PDE, a notion introduced in a recent series of papers [32, 33, 19, 16, 17].

Suggested Citation

  • Mykhaylo Shkolnikov & Lane Chun Yeung, 2024. "From rank-based models with common noise to pathwise entropy solutions of SPDEs," Papers 2406.07286, arXiv.org.
    • Handle: RePEc:arx:papers:2406.07286
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    References listed on IDEAS

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    1. Sullivan, Timothy G, 1978. "The Cost of Capital and the Market Power of Firms," The Review of Economics and Statistics, MIT Press, vol. 60(2), pages 209-217, May.
    2. Shkolnikov, Mykhaylo, 2012. "Large systems of diffusions interacting through their ranks," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1730-1747.
    3. Kolli, Praveen & Sarantsev, Andrey, 2019. "Large rank-based models with common noise," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 29-35.
    4. Gess, Benjamin & Souganidis, Panagiotis E., 2017. "Stochastic non-isotropic degenerate parabolic–hyperbolic equations," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2961-3004.
    5. Jourdain, B., 2000. "Probabilistic approximation for a porous medium equation," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 81-99, September.
    6. Binder, John J, 1992. "Beta, Firm Size, and Concentration," Economic Inquiry, Western Economic Association International, vol. 30(3), pages 556-563, July.
    7. Marti G. Subrahmanyam & Stavros B. Thomadakis, 1980. "Systematic Risk and the Theory of the Firm," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 94(3), pages 437-451.
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