Abstract
This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator of the trace-class operator-valued integrated volatility process arising in general mild solutions of Hilbert space-valued stochastic evolution equations in the sense of Da Prato and Zabczyk (2014). We prove a weak law of large numbers for this estimator, where the convergence is uniform on compacts in probability with respect to the Hilbert–Schmidt norm. In addition, we determine convergence rates for common stochastic volatility models in Hilbert spaces.
Highlights
Stochastic volatility and covariance estimation are of key importance in many fields.Motivated in particular by financial applications, a lot of research has been devoted to constructing suitable volatility estimators and to deriving their asymptotic limit theory in the setting when discrete, high-frequent observations are available
It is in line with the finite dimensional theory for continuous semimartingales that, apart from the necessary assumptions for stochastic integrability, no assumptions have to be imposed on the stochastic volatility process σ to guarantee the validity of this weak law of large numbers
We have defined the so-called semigroup-adjusted realised covariation (SARCV) and derived a weak law of large numbers based on uniform convergence in probability with respect to the Hilbert–Schmidt norm
Summary
Stochastic volatility and covariance estimation are of key importance in many fields. We prove uniform convergence in probability (ucp) with respect to the Hilbert–Schmidt norm of the (SARCV) to the integrated covariance process It is in line with the finite dimensional theory for continuous semimartingales that, apart from the necessary assumptions for stochastic integrability, no assumptions have to be imposed on the stochastic volatility process σ to guarantee the validity of this weak law of large numbers. To the best of our knowledge, our paper is the first one considering high-frequency estimation of (co-) volatility of infinite-dimensional stochastic evolution equations in an operator setting. This is of interest for various reasons.
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