Estimation of the Jump Activity Index in the Presence of Random Observation Times

This work studies the estimation of the \textit{jump activity index} of Itô semimartingales in a setting of high frequency observations with a fixed time horizon and random observation times.\\ We give a quick overview over the underlying theory and briefly review already existing literature connected to the estimation of \textit{jump activity index} in various settings.\\ We then prove a central limit theorem based on the \textit{empirical characteristic function} whose value is in our case codetermined by the (possibly unknown) structure of the underlying observation scheme. To bypass this problem we employ an approach, that is new to existing literature, using a Taylor expansion of the natural logarithm and the exponential function to develop a consistent estimator for the \textit{jump activity index}. Yet again, the connected central limit theorem (CLT) depends on the setting of the observation scheme and is therefore not directly applicable in most situations. Hence, we develop a further CLT that works without any prior knowledge of the underlying structures.