Random walks and Levy processes as rough paths

Author: 

Chevyrev, I

Publication Date: 

April 2018

Journal: 

PROBABILITY THEORY AND RELATED FIELDS

Last Updated: 

2019-05-03T07:32:20.88+01:00

Issue: 

3-4

Volume: 

170

DOI: 

10.1007/s00440-017-0781-1

page: 

891-932

abstract: 

© The Author(s) 2017. 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.

Symplectic id: 

695502

Download URL: 

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article