@phdthesis{Bossert2024,
  author    = {Bossert, Patrick},
  title     = {Statistical structure and inference methods for discrete high-frequency observations of SPDEs in one and multiple space dimensions},
  doi       = {10.25972/OPUS-36113},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-361130},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2024},
  abstract  = {The focus of this thesis is on analysing a linear stochastic partial differential equation (SPDE) with a bounded domain. The first part of the thesis commences with an examination of a one-dimensional SPDE. In this context, we construct estimators for the parameters of a parabolic SPDE based on discrete observations of a solution in time and space on a bounded domain. We establish central limit theorems for a high-frequency asymptotic regime, showing substantially smaller asymptotic variances compared to existing estimation methods. Moreover, asymptotic confidence intervals are directly feasible. Our approach builds upon realized volatilities and their asymptotic illustration as the response of a log-linear model with a spatial explanatory variable. This yields efficient estimators based on realized volatilities with optimal rates of convergence and minimal variances. We demonstrate our results by Monte Carlo simulations. Extending this framework, we analyse a second-order SPDE model in multiple space dimensions in the second part of this thesis and develop estimators for the parameters of this model based on discrete observations 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 that generalizes the theory to a general, multi-dimensional framework. Our methodology enables the construction of an oracle estimator for volatility within the underlying model. For proving central limit theorems, we use a high-frequency observation scheme. To showcase our results, we conduct a Monte Carlo simulation, highlighting the advantages of our novel approach in a multi-dimensional context.},
  subject      = {Stochastische partielle Differentialgleichung},
  language  = {en}
}