Risk bounds for the non-parametric estimation of Lévy processes

Abstract

Estimation methods for the Lévy density of a Lévy process are developed under mild qualitative assumptions. A classical model selection approach made up of two steps is studied. The first step consists in the selection of a good estimator, from an approximating (finite-dimensional) linear model $\calS$ for the true Lévy density. The second is a data-driven selection of a linear model $\calS$, among a given collection $\{\calS_{m}\}_{m\in\calM}$, that approximately realizes the best trade-off between the error of estimation within $\calS$ and the error incurred when approximating the true Lévy density by the linear model $\calS$. Using recent concentration inequalities for functionals of Poisson integrals, a bound for the risk of estimation is obtained. As a byproduct, oracle inequalities and long-run asymptotics for spline estimators are derived. Even though the resulting underlying statistics are based on continuous time observations of the process, approximations based on high-frequency discrete-data can be easily devised.