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Large volatility-stabilized markets

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  • Shkolnikov, Mykhaylo
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    Abstract

    We investigate the behavior of systems of interacting diffusion processes, known as volatility-stabilized market models in the mathematical finance literature, when the number of diffusions tends to infinity. We show that, after an appropriate rescaling of the time parameter, the empirical measure of the system converges to the solution of a degenerate parabolic partial differential equation. A stochastic representation of the latter in terms of one-dimensional distributions of a time-changed squared Bessel process allows us to give an explicit description of the limit.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0304414912001949
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    Bibliographic Info

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

    Volume (Year): 123 (2013)
    Issue (Month): 1 ()
    Pages: 212-228

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    Handle: RePEc:eee:spapps:v:123:y:2013:i:1:p:212-228

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    Related research

    Keywords: Interacting diffusion processes; Hydrodynamic limit; Volatility-stabilized models; Bessel processes; Degenerate parabolic partial differential equations;

    References

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    1. Robert Fernholz & Ioannis Karatzas, 2005. "Relative arbitrage in volatility-stabilized markets," Annals of Finance, Springer, vol. 1(2), pages 149-177, November.
    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. Adrian Banner & Daniel Fernholz, 2008. "Short-term relative arbitrage in volatility-stabilized markets," Annals of Finance, Springer, vol. 4(4), pages 445-454, October.
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