Second-order asymptotic expansion for the covariance estimator of two asynchronously observed diffusion processes
Résumé
In this paper, we study the asymptotic properties of the Hayashi-Yoshida estimator, hereafter HY-estimator, of two diffusion processes when observations are subject to non-synchronicity. Our setup includes random sampling schemes, provided that the observation times are independent of the underlying diffusions. We first derive second-order asymptotic expansions for the distribution of the Hayashi-Yoshida estimator in the case when observed diffusions have no drift. We then focus on the drifted case and carry out a stochastic decomposition of the HY-estimator itself. This decomposition, in conjunction with the evaluation of the Malliavin covariance, leads to a second-order asymptotic expansion of the distribution of the HY-estimator. This result lies in continuity of the consistency and the asymptotic normality results proved by Hayashi and Yoshida. We compute the constants involved in the obtained expansions for the particular case where the sampling scheme is generated by two independent Poisson processes.
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