Random walks and Lévy processes as rough paths

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We consider random walks and Lévy processes in a homogeneous group G. For all p>0, we completely characterise (almost) all G-valued Lévy processes whose sample paths have finite p-variation, and give sufficient conditions under which a sequence of G-valued random walks converges in law to a Lévy process in p-variation topology. In the case that G is the free nilpotent Lie group over Rd, so that processes of finite p-variation are identified with rough paths, we demonstrate applications of our results to weak convergence of stochastic flows and provide a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process. At the heart of our analysis is a criterion for tightness of p-variation for a collection of càdlàg strong Markov processes.
Original languageEnglish
Pages (from-to)891-932
Number of pages42
JournalProbability theory and related fields
Issue number3-4
Publication statusPublished - 17 May 2017


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