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：本条为在无限维度环境下的波动估计提供了一个无症状的理论。 我们考虑了半线性随机部分差异方程的温和解决方案,并为经过调整的半群体已实现的共变(SARCV)得出一个稳定的中央限值,该半群体已实现的共变(SARCV)是综合波动和对Hilbert空间已实现的二次变异的概观的一致估计。此外,我们引入了半组经调整的多功率变异(SAMPV),并建立了其大量弱定律;我们使用SAMPV, 构建了一个对SARCV中央限值中出现的混合-Gausian限制进程无症状共变的一致估计器,从而形成了一个可行的反常理论。 最后,我们概述了我们的结果如何应用,即使观测只能在离散空间时间网格上进行,我们也可以应用我们的成果。</s>