Second-order asymptotic expansion for a non-synchronous covariation estimator

TL;DR

This paper derives second-order asymptotic expansions for the Hayashi-Yoshida estimator of covariation between two diffusion processes under non-synchronous observations, enhancing understanding of its distribution in complex sampling scenarios.

Contribution

It provides a novel second-order asymptotic expansion for the estimator's distribution, accommodating general sampling schemes and non-anticipative drifts, extending previous first-order results.

Findings

Derived second-order asymptotic expansions for the estimator's distribution.

Applied the results to Poisson sampling schemes with explicit constants.

Enhanced accuracy of covariation estimation in non-synchronous data contexts.

Abstract

In this paper, we consider the problem of estimating the covariation of two diffusion processes when observations are subject to non-synchronicity. Building on recent papers \cite{Hay-Yos03, Hay-Yos04}, we derive second-order asymptotic expansions for the distribution of the Hayashi-Yoshida estimator in a fairly general setup including random sampling schemes and non-anticipative random drifts. The key steps leading to our results are a second-order decomposition of the estimator's distribution in the Gaussian set-up, a stochastic decomposition of the estimator itself and an accurate evaluation of the Malliavin covariance. To give a concrete example, we compute the constants involved in the resulting expansions for the particular case of sampling scheme generated by two independent Poisson processes.