Limit theorems for integrated local empirical characteristic exponents from noisy high-frequency data with application to volatility and jump activity estimation

We derive limit theorems for functionals of local empirical characteristic functions constructed from high-frequency observations of Itô semimartingales contaminated with noise. In a first step, we average locally the data to mitigate the effect of the noise, and then in a second step, we form local empirical characteristic functions from the pre-averaged data. The final statistics are formed by summing the local empirical characteristic exponents over the observation interval. The limit behavior of the statistics is governed by the observation noise, the diffusion coefficient of the Itô semimartingale and the behavior of its jump compensator around zero. Different choices for the block sizes for pre-averaging and formation of the local empirical characteristic function as well as for the argument of the characteristic function make the asymptotic role of the diffusion, the jumps and the noise differ. The derived limit results can be used in a wide range of applications and in particular for doing the following in a noisy setting: (1) efficient estimation of the time-integrated diffusion coefficient in presence of jumps of arbitrary activity, and (2) efficient estimation of the jump activity (Blumenthal.Getoor) index.