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.

翻译：我们分析了多空间维度下的二阶随机偏微分方程（SPDE）模型，并基于有界域内解在时间和空间上的离散观测，发展了该模型参数的估计方法。尽管近年来的文献已建立一维和二维空间情形的参数估计，但本文是首次将理论推广至通用的多维框架。我们的方法基于实现波动率，能够为底层模型中的波动率构建一个预言估计量。此外，我们证明实现波动率在渐近意义上可表示为带有空间解释变量的对数线性模型的响应。这基于实现波动率构建了新颖且高效的估计量，其具有最优收敛速率和最小方差。在证明中心极限定理时，我们采用高频观测方案。为展示结果，我们进行了蒙特卡罗模拟。