Jump filtering and efficient drift estimation for L\'evy-driven SDE's

TL;DR

This paper introduces an efficient, asymptotically normal estimator for the drift parameter in Lévy-driven SDEs, using a novel jump filtering method and nonparametric recovery of the continuous component from high-frequency discrete data.

Contribution

It develops a new drift estimation technique for Lévy-driven SDEs that relaxes previous sampling conditions and handles unobserved continuous parts effectively.

Findings

Estimator is asymptotically normal under minimal jump conditions.

Relaxed sampling condition compared to previous methods.

Effective in various Lévy-driven financial models.

Abstract

The problem of drift estimation for the solution $X$ of a stochastic differential equation with L\'evy-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 $n \Delta_n^{3-\epsilon}\to 0,$ where $n$ is the number of observations and $\Delta_n$ is the maximal sampling step. This result relaxes the condition $n\Delta_n^2 \to 0$ usually required for joint estimation of drift and diffusion coefficient for SDE's 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. Finally, we illustrate the behavior of our drift estimator for a number of popular L\'evy-driven models from finance.