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：我们分析了一个多空间维数的二阶SPDE模型，并基于有界域内解的时空离散观测数据，为该模型的参数构建了估计量。尽管一维和二维空间维度的参数估计在近期文献中已有建立，但本文是首次将该理论推广到一般多维框架。我们的方法基于实现波动率，从而能够为底层模型中的波动率构造一个神谕估计量。此外，我们证明实现波动率具有以空间解释变量为自变量的对数线性模型响应的渐近表示。这产生了基于实现波动率的新型高效估计量，该估计量具有最优收敛速率和最小方差。为证明中心极限定理，我们采用高频观测方案。最后通过蒙特卡洛模拟展示结果。