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Parameter estimation for second-order SPDEs in multiple space dimensions

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  • Patrick Bossert

    (Julius-Maximilians-Universität Würzburg)

Abstract

We analyse a second-order SPDE model in multiple space dimensions and develop estimators for the parameters of this model based on discrete observations of a solution in time and space on a bounded domain. While parameter estimation for one and two spatial dimensions was established in recent literature, this is the first work which generalizes the theory to a general, multi-dimensional framework. Our approach builds upon realized volatilities, enabling the construction of an oracle estimator for volatility within the underlying model. Furthermore, we show that the realized volatilities have an asymptotic illustration as response of a log-linear model with spatial explanatory variable. This yields novel and efficient estimators based on realized volatilities with optimal rates of convergence and minimal variances. For proving central limit theorems, we use a high-frequency observation scheme. To showcase our results, we conduct a Monte Carlo simulation.

Suggested Citation

  • Patrick Bossert, 2024. "Parameter estimation for second-order SPDEs in multiple space dimensions," Statistical Inference for Stochastic Processes, Springer, vol. 27(3), pages 485-583, October.
    • Handle: RePEc:spr:sistpr:v:27:y:2024:i:3:d:10.1007_s11203-024-09318-1
    DOI: 10.1007/s11203-024-09318-1
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    References listed on IDEAS

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    1. 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.
    2. Mena, H. & Pfurtscheller, L., 2019. "An efficient SPDE approach for El Niño," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 146-156.
    3. Hildebrandt, Florian, 2020. "On generating fully discrete samples of the stochastic heat equation on an interval," Statistics & Probability Letters, Elsevier, vol. 162(C).
    Full references (including those not matched with items on IDEAS)

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