Several Itô formulas have been already established for Lévy processes. We explain according to which criteria they are not optimal and establish an extended Itô formula that satisfies that criteria. The interest, in particular, of this formula is to obtain the explicit decomposition of $F(X)$, for $X$ Lévy process and $F$ deterministic function with locally bounded first order Radon-Nikodym derivatives, as the sum of a Dirichlet process and a bounded variation process.
"An optimal Itô formula for Lévy processes." Electron. Commun. Probab. 14 202 - 209, 2009. https://doi.org/10.1214/ECP.v14-1469