Global estimation of realized spot volatility in the presence of price jumps

We propose a non-parametric procedure for estimating the realized spot volatility of a price process described by an Itô semimartingale with Lévy jumps. The procedure integrates the threshold jump elimination technique of Mancini (2009) with a frame (Gabor) expansion of the realized trajectory of spot volatility. We show that the procedure converges in probability in L2([0, T]) for a wide class of spot volatility processes, including those with discontinuous paths. Our analysis assumes the time interval between price observations tends to zero; as a result, the intended application is for the analysis of high frequency financial data.