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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.

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Bibliographic Info

Article provided by Elsevier in its journal Stochastic Processes and their Applications.

Volume (Year): 122 (2012)
Issue (Month): 3 ()
Pages: 1068-1092

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Handle: RePEc:eee:spapps:v:122:y:2012:i:3:p:1068-1092

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Related research

Keywords: Itô processes; Discrete time observations; Change point estimation; Volatility;

References

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  1. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, vol. 53(2), pages 385-407, March.
  2. 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, 02.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  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 & 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, vol. 58(2), pages 211-222, June.
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