  In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was
introduced for a large family of Levy processes that is based on the
Wiener-Hopf decomposition. We pursue this idea further by combining their
technique with the recently introduced multilevel Monte Carlo methodology.
Moreover, we provide here for the first time a theoretical analysis of the new
Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its
multilevel variant for computing expectations of functions depending on the
historical trajectory of a Levy process. We derive rates of convergence for
both methods and show that they are uniform with respect to the "jump activity"
(e.g. characterised by the Blumenthal-Getoor index). We also present a modified
version of the algorithm in Kuznetsov et al. (2011) which combined with the
multilevel methodology obtains the optimal rate of convergence for general Levy
processes and Lipschitz functionals. This final result is only a theoretical
one at present, since it requires independent sampling from a triple of
distributions which is currently only possible for a limited number of
processes.
