Jump filtering and efficient drift estimation for Lévy-driven SDEs

Abstract

The problem of drift estimation for the solution $X$ of a stochastic differential equation with Lévy-type jumps is considered under discrete high-frequency observations with a growing observation window. An efficient and asymptotically normal estimator for the drift parameter is constructed under minimal conditions on the jump behavior and the sampling scheme. In the case of a bounded jump measure density, these conditions reduce to $n\Delta_{n}^{3-\varepsilon}\rightarrow 0$, where $n$ is the number of observations and $\Delta_{n}$ is the maximal sampling step. This result relaxes the condition $n\Delta_{n}^{2}\rightarrow 0$ usually required for joint estimation of drift and diffusion coefficient for SDEs with jumps. The main challenge in this estimation problem stems from the appearance of the unobserved continuous part $X^{c}$ in the likelihood function. In order to construct the drift estimator, we recover this continuous part from discrete observations. More precisely, we estimate, in a nonparametric way, stochastic integrals with respect to $X^{c}$. Convergence results of independent interest are proved for these nonparametric estimators.