High-frequency asymptotics for path-dependent functionals of Itô semimartingales

The estimation of local characteristics of Itô semimartingales has received a great deal of attention in both academia and industry over the past decades. In various papers limit theorems were derived for functionals of increments and ranges in the infill asymptotics setting. In this paper we establish the asymptotic theory for a wide class of statistics that are built from the incremental process of an Itô semimartingale. More specifically, we will show the law of large numbers and the associated stable central limit theorem for the path dependent functionals in the continuous setting, and discuss the asymptotic theory for range-based statistics in the discontinuous framework. Some examples from economics and physics demonstrate the potential applicability of our theoretical results in practice.