We construct estimators for the parameters of a parabolic SPDE with one spatial dimension 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. The asymptotic variances are shown to be substantially smaller compared to existing estimation methods. Moreover, asymptotic confidence intervals are directly feasible. Our approach builds upon realized volatilities and their asymptotic illustration as response of a log-linear model with spatial explanatory variable. This yields efficient estimators based on realized volatilities with optimal rates of convergence and minimal variances. We demonstrate efficiency gains compared to previous estimation methods numerically and in Monte Carlo simulations.

翻译：我们基于有界区域上时空离散观测数据，构建了单空间维度抛物型随机偏微分方程的参数估计量。在高频渐近框架下建立了中心极限定理，并证明该估计量的渐近方差显著小于现有估计方法。此外，可直接构造渐近置信区间。我们的方法以实现波动率为基础，并利用其作为含空间解释变量的对数线性模型响应的渐近表达特性。由此推导出基于实现波动率的高效估计量，其具有最优收敛速率与最小方差。通过数值实验与蒙特卡洛模拟，我们证明了该方法相较于现有估计方法的效率提升。