On non-parametric estimation of the Lévy kernel of Markov processes

We consider a recurrent Markov process which is an Itô semi-martingale. The Lévy kernel describes the law of its jumps. Based on observations X0,XΔ,…,XnΔX0,XΔ,…,XnΔ, we construct an estimator for the Lévy kernel’s density. We prove its consistency (as nΔ→∞nΔ→∞ and Δ→0Δ→0) and a central limit theorem. In the positive recurrent case, our estimator is asymptotically normal; in the null recurrent case, it is asymptotically mixed normal. Our estimator’s rate of convergence equals the non-parametric minimax rate of smooth density estimation. The asymptotic bias and variance are analogous to those of the classical Nadaraya–Watson estimator for conditional densities. Asymptotic confidence intervals are provided.