A weak law of large numbers for realised covariation in a Hilbert space setting

TL;DR

This paper extends the concept of realised covariation to Hilbert-space-valued stochastic processes, providing an estimator for the integrated volatility operator and proving a weak law of large numbers for it.

Contribution

It introduces a novel estimator for the trace-class operator-valued integrated volatility in Hilbert spaces and establishes its convergence properties.

Findings

The estimator converges uniformly on compacts in probability.

Conditions on volatility are applicable to common Hilbert space stochastic volatility models.

The approach generalizes realised covariation to infinite-dimensional settings.

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 show that the conditions on the volatility process are valid for most common stochastic volatility models in Hilbert spaces.