Risk bounds for the non-parametric estimation of L\'{e}vy processes

Estimation methods for the L\'{e}vy density of a L\'{e}vy 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 ${\mathcal{S}}$ for the true L\'{e}vy density. The second is a data-driven selection of a linear model ${\mathcal{S}}$, among a given collection $\{{\mathcal{S}}_m\}_{m\in {\mathcal{M}}}$, that approximately realizes the best trade-off between the error of estimation within ${\mathcal{S}}$ and the error incurred when approximating the true L\'{e}vy density by the linear model ${\mathcal{S}}$. 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.