IDEAS home Printed from https://ideas.repec.org/a/adr/anecst/y2007i85p115-130.html
   My bibliography  Save this article

Diffusion Processes with Polynomial Eigenfunctions

Author

Listed:
  • Christian Gouriéroux
  • Eric Renault
  • Pascale Valery

Abstract

The aim of this paper is to characterize the one-dimensional stochastic differential equations, for which the eigenfunctions of the infinitesimal generator are polynomials in y. Affine transformations of the Ornstein-Uhlenbeck process, the Cox-Ingersoll-Ross process and the Jacobi process belong to the solutions of this stochastic differential equation family. Such processes exhibit specific patterns of the drift and volatility functions and can be represented by means of a basis of polynomial transforms which can be used to approximate the likelihood function. We also discuss the constraints on parameters to ensure the nonnegativity of the volatility function and the stationarity of the process. The possibility to fully characterize the dynamic properties of these processes explain why they are benchmark models for unconstrained variables such as asset returns (Ornstein-Uhlenbeck), for nonnegative variables as volatilities or interest rates (Cox, Ingersoll, Ross), or for variables which can be interpreted as probabilities (Jacobi).

Suggested Citation

  • Christian Gouriéroux & Eric Renault & Pascale Valery, 2007. "Diffusion Processes with Polynomial Eigenfunctions," Annals of Economics and Statistics, GENES, issue 85, pages 115-130.
  • Handle: RePEc:adr:anecst:y:2007:i:85:p:115-130
    as

    Download full text from publisher

    File URL: http://www.jstor.org/stable/20079182
    Download Restriction: no

    References listed on IDEAS

    as
    1. Duffie, Darrell & Singleton, Kenneth J, 1997. " An Econometric Model of the Term Structure of Interest-Rate Swap Yields," Journal of Finance, American Finance Association, vol. 52(4), pages 1287-1321, September.
    2. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    3. Xiaohong Chen & Lars Peter Hansen & Jos´e A. Scheinkman, 2005. "Principal Components and the Long Run," Levine's Bibliography 122247000000000997, UCLA Department of Economics.
    4. Gourieroux, Christian & Jasiak, Joann, 2006. "Multivariate Jacobi process with application to smooth transitions," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 475-505.
    5. Kristian Stegenborg Larsen & Michael Sørensen, 2007. "Diffusion Models For Exchange Rates In A Target Zone," Mathematical Finance, Wiley Blackwell, vol. 17(2), pages 285-306.
    6. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters,in: Theory Of Valuation, chapter 5, pages 129-164 World Scientific Publishing Co. Pte. Ltd..
    7. Darrell Duffie & Rui Kan, 1996. "A Yield-Factor Model Of Interest Rates," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 379-406.
    8. Hansen, Lars Peter & Alexandre Scheinkman, Jose & Touzi, Nizar, 1998. "Spectral methods for identifying scalar diffusions," Journal of Econometrics, Elsevier, vol. 86(1), pages 1-32, June.
    9. Darolles, Serge & Florens, Jean-Pierre & Gourieroux, Christian, 2004. "Kernel-based nonlinear canonical analysis and time reversibility," Journal of Econometrics, Elsevier, vol. 119(2), pages 323-353, April.
    10. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(04), pages 627-627, November.
    11. Zengjing Chen & Larry Epstein, 2002. "Ambiguity, Risk, and Asset Returns in Continuous Time," Econometrica, Econometric Society, vol. 70(4), pages 1403-1443, July.
    12. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    13. Marco Cagetti & Lars Peter Hansen & Thomas Sargent & Noah Williams, 2002. "Robustness and Pricing with Uncertain Growth," Review of Financial Studies, Society for Financial Studies, vol. 15(2), pages 363-404, March.
    14. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "An Intertemporal General Equilibrium Model of Asset Prices," Econometrica, Econometric Society, vol. 53(2), pages 363-384, March.
    15. Conley, Timothy G, et al, 1997. "Short-Term Interest Rates as Subordinated Diffusions," Review of Financial Studies, Society for Financial Studies, vol. 10(3), pages 525-577.
    16. Florens, Jean-Pierre & Renault, Eric & Touzi, Nizar, 1998. "Testing For Embeddability By Stationary Reversible Continuous-Time Markov Processes," Econometric Theory, Cambridge University Press, vol. 14(06), pages 744-769, December.
    17. Pagan, A.R. & Hall, A.D. & Martin, V., 1995. "Modelling the Term Structure," Papers 284, Australian National University - Department of Economics.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Statistics

    Access and download statistics

    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:adr:anecst:y:2007:i:85:p:115-130. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Laurent Linnemer). General contact details of provider: http://edirc.repec.org/data/ensaefr.html .

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.