Estimation of state-dependent jump activity and drift for Markovian semimartingales

Abstract

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.

References in corpus (7)

• Estimating the degree of activity of jumps in high frequency data
• Efficient estimation of integrated volatility in presence of infinite variation jumps
• A remark on the rates of convergence for integrated volatility estimation in the presence of jumps
• Jump activity estimation for pure-jump semimartingales via self-normalized statistics
• Identifying the successive Blumenthal-Getoor indices of a discretely observed process
• Near-optimal estimation of jump activity in semimartingales
• On Non-parametric Estimation of the Lévy Kernel of Markov Processes