In recent years several techniques for simulating purely discontinuous Lévy processes have been developed. Due to the "infinite activity" character of most Lévy processes the achievement of satisfactory approximations is not a trivial issue. By means of two examples, one related to an optimal storage problem and the other to Bayesian nonparametric inference, the behavior of the so-called inverse Lévy measure algorithm is studied. Some hints for overcoming possible difficulties are given.
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