IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v128y2018i10p3253-3272.html
   My bibliography  Save this article

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

Author

Listed:
  • Duc, Luu Hoang
  • Tran, Tat Dat
  • Jost, Jürgen

Abstract

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain D of the phase space R and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics.

Suggested Citation

  • Duc, Luu Hoang & Tran, Tat Dat & Jost, Jürgen, 2018. "Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3253-3272.
  • Handle: RePEc:eee:spapps:v:128:y:2018:i:10:p:3253-3272
    DOI: 10.1016/j.spa.2017.10.014
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414917302740
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.spa.2017.10.014?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. Ait-Sahalia, Yacine, 1996. "Testing Continuous-Time Models of the Spot Interest Rate," The Review of Financial Studies, Society for Financial Studies, vol. 9(2), pages 385-426.
    2. 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: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    3. Deelstra, G. & Delbaen, F., 1995. "Long-term returns in stochastic interest rate models," Insurance: Mathematics and Economics, Elsevier, vol. 17(2), pages 163-169, October.
    4. Griselda Deelstra, 2000. "Long-term returns in stochastic interest rate models: applications," ULB Institutional Repository 2013/7590, ULB -- Universite Libre de Bruxelles.
    5. Griselda Deelstra & Freddy Delbaen, 1995. "Long-term returns in stochastic interest rate models," ULB Institutional Repository 2013/7578, ULB -- Universite Libre de Bruxelles.
    6. L. Bertini & L. Passalacqua, 2008. "Modelling interest rates by correlated multi-factor CIR-like processes," Papers 0807.3898, arXiv.org.
    7. Deelstra, Griselda, 2000. "Long-Term Returns in Stochastic Interest Rate Models: Applications," ASTIN Bulletin, Cambridge University Press, vol. 30(1), pages 123-140, May.
    8. Griselda Deelstra & Freddy Delbaen, 1995. "Long-term returns in stochastic interest rate models: convergence in law," ULB Institutional Repository 2013/7580, ULB -- Universite Libre de Bruxelles.
    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. Zhao, Juan, 2009. "Long time behaviour of stochastic interest rate models," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 459-463, June.
    2. Jan de Kort, 2018. "A note on the long rate in factor models of the term structure," Mathematical Finance, Wiley Blackwell, vol. 28(2), pages 656-667, April.
    3. Zhang, Zhenzhong & Tong, Jinying & Hu, Liangjian, 2016. "Long-term behavior of stochastic interest rate models with Markov switching," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 320-326.
    4. Bao, Jianhai & Yuan, Chenggui, 2013. "Long-term behavior of stochastic interest rate models with jumps and memory," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 266-272.
    5. Federico Flore & Giovanna Nappo, 2018. "A Feynman-Kac type formula for a fixed delay CIR model," Papers 1806.00997, arXiv.org.
    6. Griselda Deelstra, 2000. "Long-term returns in stochastic interest rate models: applications," ULB Institutional Repository 2013/7590, ULB -- Universite Libre de Bruxelles.
    7. Rogers, L. C. G. & Stummer, Wolfgang, 2000. "Consistent fitting of one-factor models to interest rate data," Insurance: Mathematics and Economics, Elsevier, vol. 27(1), pages 45-63, August.
    8. Gabriel Faraud & Stéphane Goutte, 2014. "Bessel Bridges Decomposition with Varying Dimension: Applications to Finance," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1375-1403, December.
    9. David Markantonis & G.-Fivos Sargentis & Panayiotis Dimitriadis & Theano Iliopoulou & Aimilia Siganou & Konstantina Moraiti & Maria Nikolinakou & Ilias Taygetos Meletopoulos & Nikos Mamassis & Demetri, 2023. "Stochastic Evaluation of the Investment Risk by the Scale of Water Infrastructures—Case Study: The Municipality of West Mani (Greece)," World, MDPI, vol. 4(1), pages 1-20, January.
    10. Agić-Šabeta Elma, 2016. "Constant Proportion Portfolio Insurance Strategy in Southeast European Markets," Business Systems Research, Sciendo, vol. 7(1), pages 59-80, March.
    11. Gabriel Faraud & Stéphane Goutte, 2012. "Bessel bridges decomposition with varying dimension. Applications to finance," Working Papers hal-00694126, HAL.
    12. Patrick Saart & Jiti Gao & Nam Hyun Kim, 2014. "Semiparametric methods in nonlinear time series analysis: a selective review," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(1), pages 141-169, March.
    13. Chen, Bin & Song, Zhaogang, 2013. "Testing whether the underlying continuous-time process follows a diffusion: An infinitesimal operator-based approach," Journal of Econometrics, Elsevier, vol. 173(1), pages 83-107.
    14. Gospodinov, Nikolay & Otsu, Taisuke, 2012. "Local GMM estimation of time series models with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 170(2), pages 476-490.
    15. Kozicki, Sharon & Tinsley, P. A., 2001. "Shifting endpoints in the term structure of interest rates," Journal of Monetary Economics, Elsevier, vol. 47(3), pages 613-652, June.
    16. Corradi, Valentina & Swanson, Norman R., 2005. "Bootstrap specification tests for diffusion processes," Journal of Econometrics, Elsevier, vol. 124(1), pages 117-148, January.
    17. Gómez-Valle, Lourdes & Marti­nez-Rodri­guez, Julia, 2008. "Modelling the term structure of interest rates: An efficient nonparametric approach," Journal of Banking & Finance, Elsevier, vol. 32(4), pages 614-623, April.
    18. Aït-Sahalia, Yacine & Park, Joon Y., 2012. "Stationarity-based specification tests for diffusions when the process is nonstationary," Journal of Econometrics, Elsevier, vol. 169(2), pages 279-292.
    19. Duffie, Darrell, 2003. "Intertemporal asset pricing theory," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, edition 1, volume 1, chapter 11, pages 639-742, Elsevier.
    20. Gil-Bazo Javier & Rubio Gonzalo, 2004. "A Nonparametric Dimension Test of the Term Structure," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 8(3), pages 1-28, September.

    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:eee:spapps:v:128:y:2018:i:10:p:3253-3272. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.