We study the parameter estimation for parabolic, linear, second-order, stochastic partial differential equations (SPDEs) observing a mild solution on a discrete grid in time and space. A high-frequency regime is considered where the mesh of the grid in the time variable goes to zero. Focusing on volatility estimation, we provide an explicit and easy to implement method of moments estimator based on squared increments. The estimator is consistent and admits a central limit theorem. This is established moreover for the joint estimation of the integrated volatility and parameters in the differential operator in a semi-parametric framework. Starting from a representation of the solution of the SPDE with Dirichlet boundary conditions as an infinite factor model and exploiting mixing-type properties of time series, the theory considerably differs from the statistics for semi-martingales literature. The performance of the method is illustrated in a simulation study.

中文翻译：

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使用高频观测值对随机偏微分方程的波动率估计

我们研究抛物线、线性、二阶、随机偏微分方程 (SPDE) 的参数估计，观察时间和空间离散网格上的温和解。考虑高频状态，其中时间变量中的网格网格变为零。专注于波动率估计，我们提供了一种基于平方增量的矩量估计器的明确且易于实现的方法。估计量是一致的，并承认中心极限定理。此外，这是为了在半参数框架中对微分算子中的综合波动率和参数进行联合估计而建立的。从用狄利克雷边界条件作为无限因子模型的 SPDE 解的表示开始，并利用时间序列的混合类型特性，该理论与半鞅文献的统计数据大不相同。在仿真研究中说明了该方法的性能。