Estimating spot volatility in the presence of infinite variation jumps

We propose a kernel estimator for the spot volatility of a semi-martingale at a given time point by using high frequency data, where the underlying process accommodates a jump part of infinite variation. The estimator is based on the representation of the characteristic function of Lévy processes. The consistency of the proposed estimator is established under some mild assumptions. By assuming that the jump part of the underlying process behaves like a symmetric stable Lévy process around 0, we establish the asymptotic normality of the proposed estimator. In particular, with a specific kernel function, the estimator is variance efficient. We conduct Monte Carlo simulation studies to assess our theoretical results and compare our estimator with existing ones.