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EM algorithm for Markov chains observed via Gaussian noise and point process information: Theory and case studies

Author

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  • Damian Camilla
  • Eksi Zehra

    (Institute for Statistics and Mathematics, Vienna University of Economics and Business, Welthandelsplatz 1, Wien, Austria)

  • Frey Rüdiger

    (Institute for Statistics and Mathematics, Vienna University of Economics and Business, Welthandelsplatz 1, Wien, Austria)

Abstract

In this paper we study parameter estimation via the Expectation Maximization (EM) algorithm for a continuous-time hidden Markov model with diffusion and point process observation. Inference problems of this type arise for instance in credit risk modelling. A key step in the application of the EM algorithm is the derivation of finite-dimensional filters for the quantities that are needed in the E-Step of the algorithm. In this context we obtain exact, unnormalized and robust filters, and we discuss their numerical implementation. Moreover, we propose several goodness-of-fit tests for hidden Markov models with Gaussian noise and point process observation. We run an extensive simulation study to test speed and accuracy of our methodology. The paper closes with an application to credit risk: we estimate the parameters of a hidden Markov model for credit quality where the observations consist of rating transitions and credit spreads for US corporations.

Suggested Citation

  • Damian Camilla & Eksi Zehra & Frey Rüdiger, 2018. "EM algorithm for Markov chains observed via Gaussian noise and point process information: Theory and case studies," Statistics & Risk Modeling, De Gruyter, vol. 35(1-2), pages 51-72, January.
    • Handle: RePEc:bpj:strimo:v:35:y:2018:i:1-2:p:51-72:n:4
    DOI: 10.1515/strm-2017-0021
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    References listed on IDEAS

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    1. Rüdiger Frey & Thorsten Schmidt, 2012. "Pricing and hedging of credit derivatives via the innovations approach to nonlinear filtering," Finance and Stochastics, Springer, vol. 16(1), pages 105-133, January.
    2. Dembo, A. & Zeitouni, O., 1986. "Parameter estimation of partially observed continuous time stochastic processes via the EM algorithm," Stochastic Processes and their Applications, Elsevier, vol. 23(1), pages 91-113, October.
    3. Rüdiger Frey & Wolfgang J. Runggaldier, 2001. "A Nonlinear Filtering Approach To Volatility Estimation With A View Towards High Frequency Data," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(02), pages 199-210.
    4. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    5. Jörn Sass & Ulrich Haussmann, 2004. "Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain," Finance and Stochastics, Springer, vol. 8(4), pages 553-577, November.
    6. Korolkiewicz, Malgorzata W. & Elliott, Robert J., 2008. "A hidden Markov model of credit quality," Journal of Economic Dynamics and Control, Elsevier, vol. 32(12), pages 3807-3819, December.
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    Cited by:

    1. Areski Cousin & Jérôme Lelong & Tom Picard, 2023. "Rating transitions forecasting: a filtering approach," Post-Print hal-03347521, HAL.

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