Asymptotic equivalence of jumps Lévy processes and their discrete counterpart

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We establish the global asymptotic equivalence between a pure jumps Lévy process {Xₜ} on the time interval [0,T] with unknown Lévy measure ν belonging to a non-parametric class and the observation of 2m² Poisson independent random variables with parameters linked with the Lévy measure ν. The equivalence result is asymptotic as m tends to infinity. The time T is kept fixed and the sample path is continuously observed. This result justifies the idea that, from a statistical point of view, knowing how many jumps fall into a grid of intervals gives asymptotically the same amount of information as observing {Xₜ}.