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

TL;DR

This paper develops an asymptotic theory for volatility estimation of SPDEs in infinite-dimensional spaces, introducing new estimators and establishing a feasible central limit theorem for realised covariation.

Contribution

It introduces semigroup adjusted realised covariation and multipower variations, extending volatility estimation to infinite-dimensional SPDEs with a feasible inference framework.

Findings

Established a stable central limit theorem for SARCV.

Proposed SAMPV for consistent covariance estimation.

Extended methods to discrete space-time observations.

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.