# Random walks and Lévy processes as rough paths

Chevyrev, I

April 2018

## Journal:

Probab. Theory Related Fields 170 (2018), no. 3-4, 891-932

## Last Updated:

2020-05-14T16:44:32.007+01:00

3-4

170

## DOI:

10.1007/s00440-017-0781-1

891-932

## 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.

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