We establish the global asymptotic equivalence between a pure jumps Lévy process with unknown Lévy measure $\nu$ and a sequence of independent Poisson random variables with parameters depending on $\nu$. Combining this result with the one in Brown and Low (1996), we deduce an asymptotic equivalence between an additive process with unknown drift and Lévy measure and a discrete model composed by a non-parametric regression plus some independent Poisson random variables.