Nonparametric density estimation for the small jumps of L\'evy processes

We consider the problem of estimating the density of the process associated with the small jumps of a pure jump L\'evy process, possibly of infinite variation, from discrete observations of one trajectory. The interest of such a question lies on the observation that even when the L\'evy measure is known, the density of the increments of the small jumps of the process cannot be computed in closed-form. We discuss results both from low and high frequency observations. In a low frequency setting, assuming the L\'evy density associated with the jumps larger than $\varepsilon\in (0,1]$ in absolute value is known, a spectral estimator relying on the convolution structure of the problem achieves a parametric rate of convergence with respect to the integrated $L_2$ loss, up to a logarithmic factor. In a high frequency setting, we remove the assumption on the knowledge of the L\'evy measure of the large jumps and show that the rate of convergence depends both on the sampling scheme and on the behaviour of the L\'evy measure in a neighborhood of zero. We show that the rate we find is minimax up to a logarithmic factor. An adaptive penalized procedure is studied to select the cutoff parameter. These results are extended to encompass the case where a Brownian component is present in the L\'evy process. Furthermore, we illustrate numerically the performances of our procedures.