Risk bounds for the non-parametric estimation of L\'evy processes
Abstract
Estimation methods for the L\'evy density of a L\'evy 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 S for the true L\'evy density. The second is a data-driven selection of a linear model S, among a given collection \Sm\m∈ M, that approximately realizes the best trade-off between the error of estimation within S and the error incurred when approximating the true L\'evy density by the linear model S. 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.
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