Global jump filters and quasi-likelihood analysis for volatility

We propose a new estimation scheme for estimation of the volatility parameters of a semimartingale with jumps based on a jump-detection filter. Our filter uses all of data to analyze the relative size of increments and to discriminate jumps more precisely. We construct quasi-maximum likelihood estimators and quasi-Bayesian estimators, and show limit theorems for them including $L^p$-estimates of the error and asymptotic mixed normality based on the framework of the quasi-likelihood analysis. The global jump filters do not need a restrictive condition for the distribution of the small jumps. By numerical simulation we show that our "global" method obtains better estimates of the volatility parameter than the previous "local" methods.