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Estimation for the change point of volatility in a stochastic differential equation

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  • Iacus, Stefano M.
  • Yoshida, Nakahiro

Abstract

We consider a multidimensional Itô process Y=(Yt)t∈[0,T] with some unknown drift coefficient process bt and volatility coefficient σ(Xt,θ) with covariate process X=(Xt)t∈[0,T], the function σ(x,θ) being known up to θ∈Θ. For this model, we consider a change point problem for the parameter θ in the volatility component. The change is supposed to occur at some point t∗∈(0,T). Given discrete time observations from the process (X,Y), we propose quasi-maximum likelihood estimation of the change point. We present the rate of convergence of the change point estimator and the limit theorems of the asymptotically mixed type.

Suggested Citation

  • Iacus, Stefano M. & Yoshida, Nakahiro, 2012. "Estimation for the change point of volatility in a stochastic differential equation," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 1068-1092.
  • Handle: RePEc:eee:spapps:v:122:y:2012:i:3:p:1068-1092
    DOI: 10.1016/j.spa.2011.11.005
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    References listed on IDEAS

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    1. Jushan Bai, 1997. "Estimation Of A Change Point In Multiple Regression Models," The Review of Economics and Statistics, MIT Press, vol. 79(4), pages 551-563, November.
    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: Theory Of Valuation, chapter 5, pages 129-164 World Scientific Publishing Co. Pte. Ltd..
    3. Sangyeol Lee & Yoichi Nishiyama & Nakahiro Yoshida, 2006. "Test for Parameter Change in Diffusion Processes by Cusum Statistics Based on One-step Estimators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(2), pages 211-222, June.
    4. Junmo Song & Sangyeol Lee, 2009. "Test for parameter change in discretely observed diffusion processes," Statistical Inference for Stochastic Processes, Springer, vol. 12(2), pages 165-183, June.
    5. Chen, Gongmeng & Choi, Yoon K. & Zhou, Yong, 2005. "Nonparametric estimation of structural change points in volatility models for time series," Journal of Econometrics, Elsevier, vol. 126(1), pages 79-114, May.
    6. Stefano M. Iacus & Nakahiro Yoshida, 2010. "Numerical Analysis of Volatility Change Point Estimators for Discretely Sampled Stochastic Differential Equations," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 39(s1), pages 107-127, February.
    7. Shixin, Gan, 1997. "The Hájek-Rényi inequality for Banach space valued martingales and the p smoothness of Banach spaces," Statistics & Probability Letters, Elsevier, vol. 32(3), pages 245-248, March.
    8. Sangyeol Lee & Jeongcheol Ha & Okyoung Na & Seongryong Na, 2003. "The Cusum Test for Parameter Change in Time Series Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(4), pages 781-796.
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    Cited by:

    1. Markus Bibinger & Moritz Jirak & Mathias Vetter, 2015. "Nonparametric change-point analysis of volatility," SFB 649 Discussion Papers SFB649DP2015-008, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

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