Asymptotic efficiency for covariance estimation under noise and asynchronicity

TL;DR

This paper investigates the limits of estimating covariance matrices from noisy, asynchronously observed Gaussian processes at high frequency, establishing fundamental bounds and convolution theorems relevant to finance.

Contribution

It introduces asymptotic bounds and convolution theorems for covariance estimation under noise and asynchronicity in Gaussian processes, providing benchmarks for various models.

Findings

Established asymptotic lower and upper bounds for covariance estimation.

Derived infinite-dimensional convolution theorems for covariation estimation.

Provided benchmark cases for Gaussian process models in high-frequency data.

Abstract

The estimation of the covariance structure from a discretely observed multivariate Gaussian process under asynchronicity and noise is analysed under high-frequency asymptotics. Asymptotic lower and upper bounds are established for a general Gaussian framework which provides benchmark cases for various Gaussian process models of interest. The parametric bounds give rise to infinite-dimensional convolution theorems for covariation estimation under asynchronicity, which is an essential estimation problem in finance.