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Inference for systems of stochastic differential equations from discretely sampled data: A numerical maximum likelihood approach

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  • Lux, Thomas

Abstract

Maximum likelihood estimation of discretely observed diffusion processes is mostly hampered by the lack of a closed form solution of the transient density. It has recently been argued that a most generic remedy to this problem is the numerical solution of the pertinent Fokker-Planck (FP) or forward Kol- mogorov equation. Here we expand extant work on univariate diffusions to higher dimensions. We find that in the bivariate and trivariate cases, a numerical solution of the FP equation via alternating direction finite difference schemes yields results surprisingly close to exact maximum likelihood in a number of test cases. After providing evidence for the effciency of such a numerical approach, we illustrate its application for the estimation of a joint system of short-run and medium run investor sentiment and asset price dynamics using German stock market data.

Suggested Citation

  • Lux, Thomas, 2012. "Inference for systems of stochastic differential equations from discretely sampled data: A numerical maximum likelihood approach," Kiel Working Papers 1781, Kiel Institute for the World Economy (IfW Kiel).
    • Handle: RePEc:zbw:ifwkwp:1781
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    1. Creedy, John & Lye, Jenny & Martin, Vance L, 1996. "A Non-linear Model of the Real US-UK Exchange Rate," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(6), pages 669-686, Nov.-Dec..
    2. Lux, Thomas, 2009. "Rational forecasts or social opinion dynamics? Identification of interaction effects in a business climate survey," Journal of Economic Behavior & Organization, Elsevier, vol. 72(2), pages 638-655, November.
    3. Thomas Lux, 2011. "Sentiment dynamics and stock returns: the case of the German stock market," Empirical Economics, Springer, vol. 41(3), pages 663-679, December.
    4. Yacine Aït-Sahalia, 2001. "Transition Densities For Interest Rate And Other Nonlinear Diffusions," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar(Volume II), chapter 1, pages 1-34, World Scientific Publishing Co. Pte. Ltd..
    5. A. S. Hurn & J. I. Jeisman & K. A. Lindsay, 0. "Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations," Journal of Financial Econometrics, Oxford University Press, vol. 5(3), pages 390-455.
    6. Lux, Thomas, 2012. "Estimation of an agent-based model of investor sentiment formation in financial markets," Journal of Economic Dynamics and Control, Elsevier, vol. 36(8), pages 1284-1302.
    7. Thomas Lux, 2009. "Rational Forecasts or Social Opinion Dynamics? Identification of Interaction Effects in a Business Climate Survey," Post-Print hal-00720175, HAL.
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    Cited by:

    1. esposito, francesco paolo & cummins, mark, 2015. "Filtering and likelihood estimation of latent factor jump-diffusions with an application to stochastic volatility models," MPRA Paper 64987, University Library of Munich, Germany.
    2. Dmitry Zhukov & Julia Perova & Vladimir Kalinin, 2022. "Description of the Distribution Law and Non-Linear Dynamics of Growth of Comments Number in News and Blogs Based on the Fokker-Planck Equation," Mathematics, MDPI, vol. 10(6), pages 1-24, March.

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    More about this item

    Keywords

    stochastic differential equations; numerical maximum likelihood; Fokker-Planck equation; finite difference schemes; asset pricing;
    All these keywords.

    JEL classification:

    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

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