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. 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 minimax parametric rates 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 log-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 the performances of our procedures through an extensive simulation study.

翻译：我们考虑从单条轨道的离散观测数据中，估计（可能为无穷变差）纯跳Lévy过程小跳相关过程的密度问题。该问题的研究价值在于：即使已知Lévy测度，过程小跳增量的密度仍无法直接计算。我们分别讨论低频与高频观测场景下的结论。在低频设定下，假设绝对值大于$\varepsilon\in (0,1]$的跳所对应的Lévy密度已知，基于该问题卷积结构构建的谱估计器可在积分$L_2$损失下达到极小极大参数收敛速率（仅含对数因子）。在高频设定下，我们移除了大跳Lévy测度已知的假设，并证明收敛速率同时取决于采样方案与Lévy测度在零点邻域内的行为。我们证明该速率在含对数因子意义下达到极小最优。为选择截断参数，我们研究了自适应惩罚方法。上述结果可推广至Lévy过程包含布朗运动分量的情形。此外，通过广泛的模拟研究验证了所提方法的性能。