This article establishes an asymptotic theory for volatility estimation in an infinite-dimensional setting. We consider mild solutions of semilinear stochastic partial differential equations and derive a stable central limit theorem for the semigroup adjusted realised covariation (SARCV), which is a consistent estimator of the integrated volatility and a generalisation of the realised quadratic covariation to Hilbert spaces. Moreover, we introduce semigroup adjusted multipower variations (SAMPV) and establish their weak law of large numbers; using SAMPV, we construct a consistent estimator of the asymptotic covariance of the mixed-Gaussian limiting process appearing in the central limit theorem for the SARCV, resulting in a feasible asymptotic theory. Finally, we outline how our results can be applied even if observations are only available on a discrete space-time grid.

翻译：本文建立了无穷维框架下波动率估计的渐近理论。我们考虑半线性随机偏微分方程的温和解，导出了半群调整可实现协方差(SARCV)的稳定中心极限定理，该统计量为积分波动率的一致估计量，也是Hilbert空间中可实现二次协方差的推广。此外，我们引入了半群调整多幂变差(SAMPV)并建立了其弱大数定律；利用SAMPV，我们构造了SARCV中心极限定理中出现的混合高斯极限过程渐近协方差的一致估计量，从而得到可行的渐近理论。最后，我们概述了即使在离散时空网格上仅能获取观测数据时，如何应用本文的结果。