Approximations for the distributions of bounded variation Lévy processes

We propose a feasible method for approximating the marginal distributions and densities of a bounded variation Lévy process using polynomial expansions. We provide a fast recursive formula for approximating the coefficients of the expansions and estimating the order of the approximation error. Our expansions are shown to be the exact counterpart of successive approximations of the Lévy process by compound Poisson processes previously proposed by, for instance, Barndorff-Nielsen and Hubalek (2008) [Barndorff-Nielsen, O.E., Hubalek, F., 2008. Probability measures, Lévy measures, and analyticity in time. Bernoulli 3 (14), 764–790] and others, and hence, give an answer to an open problem raised therein.