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Non parametric estimation of the diffusion coefficients of a diffusion with jumps

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  • Schmisser, Émeline
Abstract
In this article, we consider a jump diffusion process Xtt≥0, with drift function b, diffusion coefficient σ and jump coefficient ξ2. This process is observed at discrete times t=0,Δ,…,nΔ. The sampling interval Δ tends to 0 and the time interval nΔ tends to infinity. We assume that Xtt≥0 is ergodic, strictly stationary and exponentially β-mixing. We use a penalized least-square approach to compute adaptive estimators of the functions σ2+ξ2 and σ2. We provide bounds for the risks of the two estimators.

Suggested Citation

  • Schmisser, Émeline, 2019. "Non parametric estimation of the diffusion coefficients of a diffusion with jumps," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5364-5405.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:12:p:5364-5405
    DOI: 10.1016/j.spa.2019.03.003
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    References listed on IDEAS

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    1. Kappus, Johanna, 2014. "Adaptive nonparametric estimation for Lévy processes observed at low frequency," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 730-758.
    2. Mancini, Cecilia & Renò, Roberto, 2011. "Threshold estimation of Markov models with jumps and interest rate modeling," Journal of Econometrics, Elsevier, vol. 160(1), pages 77-92, January.
    3. Emeline Schmisser, 2012. "Non-parametric estimation of the diffusion coefficient from noisy data," Statistical Inference for Stochastic Processes, Springer, vol. 15(3), pages 193-223, October.
    4. Comte, F. & Rozenholc, Y., 2002. "Adaptive estimation of mean and volatility functions in (auto-)regressive models," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 111-145, January.
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    6. Arnaud Gloter & Dasha Loukianova & Hilmar Mai, 2016. "Jump filtering and efficient drift estimation for Lévy-Driven SDE’S," Working Papers 2016-04, Center for Research in Economics and Statistics.
    7. Shota Gugushvili, 2009. "Nonparametric estimation of the characteristic triplet of a discretely observed Lévy process," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(3), pages 321-343.
    8. Yasutaka Shimizu & Nakahiro Yoshida, 2006. "Estimation of Parameters for Diffusion Processes with Jumps from Discrete Observations," Statistical Inference for Stochastic Processes, Springer, vol. 9(3), pages 227-277, October.
    9. Masuda, Hiroki, 2007. "Ergodicity and exponential [beta]-mixing bounds for multidimensional diffusions with jumps," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 35-56, January.
    10. Comte, F. & Genon-Catalot, V., 2009. "Nonparametric estimation for pure jump Lévy processes based on high frequency data," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4088-4123, December.
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    Cited by:

    1. Wang, Bin & Zheng, Xu, 2022. "Testing for the presence of jump components in jump diffusion models," Journal of Econometrics, Elsevier, vol. 230(2), pages 483-509.
    2. Eddy Ella-Mintsa, 2024. "Nonparametric estimation of the diffusion coefficient from i.i.d. S.D.E. paths," Statistical Inference for Stochastic Processes, Springer, vol. 27(3), pages 585-640, October.

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