A note on the limit theorems for hitting times of path-dependent functionals of Itô semimartingales

This paper studies the asymptotic properties for the counts of successive hitting times of path-dependent functionals of continuous Itô semimartingales in a fixed time span. We prove that for a wide class of functionals, the counts of hitting times can be used to construct consistent and asymptotically normal estimators of the quadratic variation of the underlying continuous Itô semimartingale, which can achieve a sizeable variance reduction compared to the existing realized variance-type methods under a common sampling frequency.