Recovering Brownian and jump parts from high-frequency observations of a Lévy process

We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations a L\'evy process. The first procedure relies on reordering independently sampled normal increments thus avoids tuning parameters. functionality this method is consequence small time predominance component, presence exchangeable structures, fast convergence empirical quantile functions. second amounts to filtering compensating with final value. It requires carefully chosen threshold, in which case both yield same rate convergence. This depends small-jump activity given terms Blumenthal-Getoor index. Finally, we discuss possible extensions, including multidimensional case, provide numerical illustrations.