Random walks and Lévy processes as rough paths

Abstract

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 ${\mathbb{R}}^d$, 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.

36 pages. Revised from previous version. To appear in Probability Theory and Related Fields

References in corpus (2)

• Non-standard approximations of the Ito-map
• A support and density theorem for Markovian rough paths

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