An estimator for the cumulative co-volatility of asynchronously observed semimartingales with jumps

type="main" xml:id="sjos12043-abs-0001"> <title type="main">Abstract</title>In this paper, we consider two semimartingales sampled at stopping times in an asynchronous manner. We are interested in estimating their cumulative co-volatility separately from the sum of their co-jumps. For this purpose, we combine the Hayashi–Yoshida method (to deal with the asynchronicity) with the threshold technique (to separate the jumps) and consider a class of statistics called the truncated Hayashi–Yoshida estimator. We prove the consistency and the asymptotic mixed normality of the truncated Hayashi–Yoshida estimator under some mild conditions allowing the presence of infinite activity jumps.