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Parameter Estimation for a Discretely Observed Integrated Diffusion Process

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  • ARNAUD GLOTER

Abstract

. We consider the estimation of unknown parameters in the drift and diffusion coefficients of a one‐dimensional ergodic diffusion X when the observation is a discrete sampling of the integral of X at times iΔ,i = 1,…,n. Assuming that the sampling interval tends to 0 while the total length time interval tends to infinity, we first prove limit theorems for functionals associated with our observations. We apply these results to obtain a contrast function. The associated minimum contrast estimators are shown to be consistent and asymptotically Gaussian with different rates for drift and diffusion coefficient parameters.

Suggested Citation

  • Arnaud Gloter, 2006. "Parameter Estimation for a Discretely Observed Integrated Diffusion Process," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(1), pages 83-104, March.
    • Handle: RePEc:bla:scjsta:v:33:y:2006:i:1:p:83-104
    DOI: 10.1111/j.1467-9469.2006.00465.x
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    Cited by:

    1. Julie Lyng Forman & Michael Sørensen, 2008. "The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 438-465, September.
    2. Nicolau, João, 2008. "Modeling financial time series through second-order stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2700-2704, November.
    3. Samson, Adeline & Thieullen, Michèle, 2012. "A contrast estimator for completely or partially observed hypoelliptic diffusion," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2521-2552.
    4. Shu, Huisheng & Jiang, Ziwei & Zhang, Xuekang, 2023. "Parameter estimation for integrated Ornstein–Uhlenbeck processes with small Lévy noises," Statistics & Probability Letters, Elsevier, vol. 199(C).
    5. Arnaud Gloter, 2007. "Efficient estimation of drift parameters in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(4), pages 495-519, October.
    6. Comte, F. & Genon-Catalot, V. & Rozenholc, Y., 2009. "Nonparametric adaptive estimation for integrated diffusions," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 811-834, March.
    7. Jean Jacod & Mark Podolskij, 2012. "A Test for the Rank of the Volatility Process: The Random Perturbation Approach," Global COE Hi-Stat Discussion Paper Series gd12-268, Institute of Economic Research, Hitotsubashi University.
    8. Comte, Fabienne & Prieur, Clémentine & Samson, Adeline, 2017. "Adaptive estimation for stochastic damping Hamiltonian systems under partial observation," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3689-3718.
    9. Jean Jacod & Mark Podolskij, 2012. "A test for the rank of the volatility process: the random perturbation approach," CREATES Research Papers 2012-57, Department of Economics and Business Economics, Aarhus University.
    10. Song Yuping & Hou Weijie & Zhou Shengyi, 2019. "Variance reduction estimation for return models with jumps using gamma asymmetric kernels," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 23(5), pages 1-38, December.
    11. Susanne Ditlevsen & Adeline Samson, 2019. "Hypoelliptic diffusions: filtering and inference from complete and partial observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 361-384, April.
    12. Quentin Clairon & Adeline Samson, 2022. "Optimal control for parameter estimation in partially observed hypoelliptic stochastic differential equations," Computational Statistics, Springer, vol. 37(5), pages 2471-2491, November.
    13. Yunyan Wang & Lixin Zhang & Mingtian Tang, 2012. "Re-weighted functional estimation of second-order diffusion processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(8), pages 1129-1151, November.
    14. Quentin Clairon & Adeline Samson, 2020. "Optimal control for estimation in partially observed elliptic and hypoelliptic linear stochastic differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 105-127, April.
    15. Salima El Kolei & Fabien Navarro, 2022. "Contrast estimation for noisy observations of diffusion processes via closed-form density expansions," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 303-336, July.

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