Author
Chevyrev, I
Journal title
Probab. Theory Related Fields 170 (2018), no. 3-4, 891-932
DOI
10.1007/s00440-017-0781-1
Issue
3-4
Volume
170
Last updated
2020-08-11T10:08:19.707+01:00
Page
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.
Symplectic ID
695502
Download URL
http://arxiv.org/abs/1510.09066v2
Publication type
Journal Article
Publication date
April 2018
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