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

Fred Espen Benth; Dennis Schroers; Almut E. D. Veraart

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arXiv · 2205.03927

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

Fred Espen Benth·Dennis Schroers·Almut E. D. Veraart

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Abstract

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.

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Fred Espen Benth, Dennis Schroers, Almut E. D. Veraart. 2022-05-08. A feasible central limit theorem for realised covariation of SPDEs in the context of functional data. https://arxiv.org/abs/2205.03927

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