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

A feasible central limit theorem for realised covariation of SPDEs in the context of functional data