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Nonlinear Fokker-Planck Equation in the Model of Asset Returns

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  • Alexander Shapovalov
  • Andrey Trifonov
  • Elena Masalova
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    Abstract

    The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker-Planck equation we describe a construction of exact solution of the Cauchy problem. In the general case, we construct the leading term of the Cauchy problem solution asymptotic in a formal small parameter in semiclassical approximation following the complex WKB-Maslov method in the class of trajectory concentrated functions.

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    File URL: http://arxiv.org/pdf/0804.0900
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0804.0900.

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    Date of creation: Apr 2008
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    Publication status: Published in SIGMA 4 (2008), 038, 10 pages
    Handle: RePEc:arx:papers:0804.0900

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    Web page: http://arxiv.org/

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    1. J-F. Muzy & D. Sornette & J. delour & A. Arneodo, 2001. "Multifractal returns and hierarchical portfolio theory," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 1(1), pages 131-148.
    2. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, Econometric Society, vol. 50(4), pages 987-1007, July.
    3. Bollerslev, Tim & Chou, Ray Y. & Kroner, Kenneth F., 1992. "ARCH modeling in finance : A review of the theory and empirical evidence," Journal of Econometrics, Elsevier, Elsevier, vol. 52(1-2), pages 5-59.
    4. Sornette, Didier, 2001. "Fokker–Planck equation of distributions of financial returns and power laws," Physica A: Statistical Mechanics and its Applications, Elsevier, Elsevier, vol. 290(1), pages 211-217.
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