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.

翻译：我们考虑从一条轨迹的离散观测中估计纯跳跃Lévy过程（可能具有无限变差）小跳跃对应过程的密度问题。该问题的研究意义在于，即使Lévy测度已知，过程小跳跃增量的密度也无法以闭式形式计算。我们分别讨论了低频与高频观测下的结果。在低频观测设定中，假设绝对值大于$\varepsilon\in (0,1]$的跳跃所对应的Lévy密度已知，基于问题卷积结构的谱估计器在积分$L_2$损失下能达到参数收敛速率（至多相差对数因子）。在高频观测设定中，我们取消了对大跳跃Lévy测度已知的假设，证明收敛速率同时取决于采样方案和Lévy测度在零点邻域内的行为。我们证明该速率在对数因子范围内是最小最大最优的。研究了一种自适应惩罚程序来选择截断参数。这些结果被推广至Lévy过程包含布朗分量的情形。此外，我们通过数值模拟展示了所提方法的性能。