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Volatility estimation for stochastic PDEs using high-frequency observations

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  • Bibinger, Markus
  • Trabs, Mathias

Abstract

We study the parameter estimation for parabolic, linear, second-order, stochastic partial differential equations (SPDEs) observing a mild solution on a discrete grid in time and space. A high-frequency regime is considered where the mesh of the grid in the time variable goes to zero. Focusing on volatility estimation, we provide an explicit and easy to implement method of moments estimator based on squared increments. The estimator is consistent and admits a central limit theorem. This is established moreover for the joint estimation of the integrated volatility and parameters in the differential operator in a semi-parametric framework. Starting from a representation of the solution of the SPDE with Dirichlet boundary conditions as an infinite factor model and exploiting mixing-type properties of time series, the theory considerably differs from the statistics for semi-martingales literature. The performance of the method is illustrated in a simulation study.

Suggested Citation

  • Bibinger, Markus & Trabs, Mathias, 2020. "Volatility estimation for stochastic PDEs using high-frequency observations," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3005-3052.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:3005-3052
    DOI: 10.1016/j.spa.2019.09.002
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    References listed on IDEAS

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    1. Santa-Clara, Pedro & Sornette, Didier, 2001. "The Dynamics of the Forward Interest Rate Curve with Stochastic String Shocks," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 149-185.
    2. Jean-Philippe Bouchaud & Nicolas Sagna & Rama Cont & Nicole El-Karoui & Marc Potters, 1999. "Phenomenology of the interest rate curve," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 209-232.
    3. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    4. Igor Cialenco, 2018. "Statistical inference for SPDEs: an overview," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 309-329, July.
    5. Lars Winkelmann & Markus Bibinger & Tobias Linzert, 2016. "ECB Monetary Policy Surprises: Identification Through Cojumps in Interest Rates," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 31(4), pages 613-629, June.
    6. Per A. Mykland & Lan Zhang, 2009. "Inference for Continuous Semimartingales Observed at High Frequency," Econometrica, Econometric Society, vol. 77(5), pages 1403-1445, September.
    7. Cialenco, Igor & Glatt-Holtz, Nathan, 2011. "Parameter estimation for the stochastically perturbed Navier-Stokes equations," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 701-724, April.
    8. Thorsten Schmidt, 2006. "An Infinite Factor Model For Credit Risk," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(01), pages 43-68.
    9. Barndorff-Nielsen, Ole E. & Corcuera, José Manuel & Podolskij, Mark, 2009. "Power variation for Gaussian processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1845-1865, June.
    10. Igor Cialenco & Ruoting Gong & Yicong Huang, 2018. "Trajectory fitting estimators for SPDEs driven by additive noise," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 1-19, April.
    11. Rama Cont, 2005. "Modeling Term Structure Dynamics: An Infinite Dimensional Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(03), pages 357-380.
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    Cited by:

    1. Dennis Schroers, 2024. "Robust Functional Data Analysis for Stochastic Evolution Equations in Infinite Dimensions," Papers 2401.16286, arXiv.org, revised Jun 2024.
    2. Benth, Fred Espen & Schroers, Dennis & Veraart, Almut E.D., 2022. "A weak law of large numbers for realised covariation in a Hilbert space setting," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 241-268.
    3. Cialenco, Igor & Kim, Hyun-Jung, 2022. "Parameter estimation for discretely sampled stochastic heat equation driven by space-only noise," Stochastic Processes and their Applications, Elsevier, vol. 143(C), pages 1-30.
    4. Hildebrandt, Florian & Trabs, Mathias, 2023. "Nonparametric calibration for stochastic reaction–diffusion equations based on discrete observations," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 171-217.

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