Second-order asymptotic expansion for the covariance estimator of two asynchronously observed diffusion processes

Abstract. 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 [12, 13]. We compute the constants involved in the obtained expansions for the particular case where the sampling scheme is generated by two independent Poisson processes.