IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v24y2011i4d10.1007_s10959-010-0316-5.html
   My bibliography  Save this article

On the Imbedding Problem for Three-State Time Homogeneous Markov Chains with Coinciding Negative Eigenvalues

Author

Listed:
  • Yong Chen

    (Hunan University of Science and Technology)

  • Jianmin Chen

    (Hunan University of Science and Technology)

Abstract

For an indecomposable 3×3 stochastic matrix (i.e., 1-step transition probability matrix) with coinciding negative eigenvalues, a new necessary and sufficient condition of the imbedding problem for time homogeneous Markov chains is shown by means of an alternate parameterization of the transition rate matrix (i.e., intensity matrix, infinitesimal generator), which avoids calculating matrix logarithm or matrix square root. In addition, an implicit description of the imbedding problem for the 3×3 stochastic matrix in Johansen (J. Lond. Math. Soc. 8:345–351, 1974) is pointed out.

Suggested Citation

  • Yong Chen & Jianmin Chen, 2011. "On the Imbedding Problem for Three-State Time Homogeneous Markov Chains with Coinciding Negative Eigenvalues," Journal of Theoretical Probability, Springer, vol. 24(4), pages 928-938, December.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:4:d:10.1007_s10959-010-0316-5
    DOI: 10.1007/s10959-010-0316-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-010-0316-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-010-0316-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. McCausland, William J., 2007. "Time reversibility of stationary regular finite-state Markov chains," Journal of Econometrics, Elsevier, vol. 136(1), pages 303-318, January.
    2. Mogens Bladt & Michael Sørensen, 2005. "Statistical inference for discretely observed Markov jump processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 395-410, June.
    3. Robert B. Israel & Jeffrey S. Rosenthal & Jason Z. Wei, 2001. "Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 245-265, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Linda Möstel & Marius Pfeuffer & Matthias Fischer, 2020. "Statistical inference for Markov chains with applications to credit risk," Computational Statistics, Springer, vol. 35(4), pages 1659-1684, December.
    2. Alan Riva-Palacio & Ramsés H. Mena & Stephen G. Walker, 2023. "On the estimation of partially observed continuous-time Markov chains," Computational Statistics, Springer, vol. 38(3), pages 1357-1389, September.
    3. Yasunari Inamura, 2006. "Estimating Continuous Time Transition Matrices From Discretely Observed Data," Bank of Japan Working Paper Series 06-E-7, Bank of Japan.
    4. Jia, Chen, 2016. "A solution to the reversible embedding problem for finite Markov chains," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 122-130.
    5. Greig Smith & Goncalo dos Reis, 2017. "Robust and Consistent Estimation of Generators in Credit Risk," Papers 1702.08867, arXiv.org, revised Oct 2017.
    6. P. Lencastre & F. Raischel & P. G. Lind, 2014. "The effect of the number of states on the validity of credit ratings," Papers 1409.2661, arXiv.org.
    7. Jolakoski, Petar & Pal, Arnab & Sandev, Trifce & Kocarev, Ljupco & Metzler, Ralf & Stojkoski, Viktor, 2023. "A first passage under resetting approach to income dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    8. Zhou, Richard, 2010. "Counterparty Risk Subject To ATE," MPRA Paper 28067, University Library of Munich, Germany.
    9. Guglielmo D'Amico & Riccardo De Blasis & Philippe Regnault, 2020. "Confidence sets for dynamic poverty indexes," Papers 2006.06595, arXiv.org.
    10. Georges Dionne & Geneviève Gauthier & Khemais Hammami & Mathieu Maurice & Jean‐Guy Simonato, 2010. "Default Risk in Corporate Yield Spreads," Financial Management, Financial Management Association International, vol. 39(2), pages 707-731, June.
    11. McCausland, William J., 2007. "Time reversibility of stationary regular finite-state Markov chains," Journal of Econometrics, Elsevier, vol. 136(1), pages 303-318, January.
    12. Blöchlinger, Andreas, 2011. "Arbitrage-free credit pricing using default probabilities and risk sensitivities," Journal of Banking & Finance, Elsevier, vol. 35(2), pages 268-281, February.
    13. Beare, Brendan K. & Seo, Juwon, 2014. "Time Irreversible Copula-Based Markov Models," Econometric Theory, Cambridge University Press, vol. 30(5), pages 923-960, October.
    14. Areski Cousin & Mohamed Reda Kheliouen, 2016. "A comparative study on the estimation of factor migration models," Working Papers halshs-01351926, HAL.
    15. Mogens Bladt & Michael SØrensen, 2009. "Efficient estimation of transition rates between credit ratings from observations at discrete time points," Quantitative Finance, Taylor & Francis Journals, vol. 9(2), pages 147-160.
    16. Tamás Kristóf, 2021. "Sovereign Default Forecasting in the Era of the COVID-19 Crisis," JRFM, MDPI, vol. 14(10), pages 1-24, October.
    17. Pedro Lencastre & Frank Raischel & Pedro G. Lind & Tim Rogers, 2014. "Are credit ratings time-homogeneous and Markov?," Papers 1403.8018, arXiv.org, revised Oct 2014.
    18. Sascha Wilkens & Jean†Baptiste Brunac & Vladimir Chorniy, 2013. "IRC and CRM: Modelling Framework for the ‘Basel 2.5’ Risk Measures," European Financial Management, European Financial Management Association, vol. 19(4), pages 801-829, September.
    19. Barsotti, Flavia & De Castro, Yohann & Espinasse, Thibault & Rochet, Paul, 2014. "Estimating the transition matrix of a Markov chain observed at random times," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 98-105.
    20. Jeong, Minsoo, 2022. "Modelling persistent stationary processes in continuous time," Economic Modelling, Elsevier, vol. 109(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:24:y:2011:i:4:d:10.1007_s10959-010-0316-5. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.