Random walks and L\'evy processes as rough paths

TL;DR

This paper characterizes G-valued Lévy processes with finite p-variation, establishes convergence conditions for random walks to Lévy processes in p-variation topology, and applies these results to rough paths and stochastic flows.

Contribution

It provides a complete characterization of G-valued Lévy processes with finite p-variation and introduces new convergence criteria for random walks in this setting.

Findings

Characterization of Lévy processes with finite p-variation in homogeneous groups

Sufficient conditions for convergence of G-valued random walks to Lévy processes

Application to rough paths and stochastic flow convergence

Abstract

We consider random walks and L\'evy processes in a homogeneous group $G$. For all $p > 0$, we completely characterise (almost) all $G$-valued L\'evy 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 L\'evy 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 L\'evy-Khintchine formula for the characteristic function of the signature of a L\'evy process. At the heart of our analysis is a criterion for tightness of $p$-variation for a collection of c\`adl\`ag strong Markov processes.