Random walks and Lévy processes as rough paths

Abstract: We consider random walks and Lévy processes in the free nilpotent Lie group as rough paths. For any p > 1, we completely characterise (almost) all Lévy processes whose sample paths have finite p-variation, provide a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process treated as a rough path, and give sufficient conditions under which a sequence of random walks converges weakly to a Lévy process in rough path topologies. At the heart of our analysis is a criterion for tightness of p-variation for a collection of càdlàg strong Markov processes. We demonstrate applications of our results to weak convergence of stochastic flows.