The jump behavior of an infinitely active Itô semimartingale can be conveniently characterized by a jump activity index of Blumenthal-Getoor type, typically assumed to be constant in time. We study Markovian semimartingales with a non-constant, state-dependent jump activity index and a non-vanishing continuous diffusion component. A nonparametric estimator for the functional jump activity index is proposed and shown to be asymptotically normal under combined high-frequency and long-time-span asymptotics. Furthermore, we propose a nonparametric drift estimator which is robust to symmetric jumps of infinite variance and infinite variation, and which attains the same asymptotic variance as for a continuous diffusion process. Simulations demonstrate the finite sample behavior of our proposed estimators. The mathematical results are based on a novel uniform bound on the Markov generator of the jump diffusion. 

Keywords:  infinite activity; drift estimation; nonparametric inference; high-frequency asymptotics; infinite variance