Random walks and Lévy processes as rough paths

Random walks and Lévy processes as rough paths We consider random walks and Lévy processes in a homogeneous group G. For all $$p > 0$$ 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 $$\mathbb {R}^d$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Probability Theory and Related Fields Springer Journals

Random walks and Lévy processes as rough paths

Loading next page...
 
/lp/springer_journal/random-walks-and-l-vy-processes-as-rough-paths-ARof7j03oZ
Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by The Author(s)
Subject
Mathematics; Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathematical Finance/Insurance; Operations Research/Decision Theory
ISSN
0178-8051
eISSN
1432-2064
D.O.I.
10.1007/s00440-017-0781-1
Publisher site
See Article on Publisher Site

Abstract

We consider random walks and Lévy processes in a homogeneous group G. For all $$p > 0$$ 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 $$\mathbb {R}^d$$ 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.

Journal

Probability Theory and Related FieldsSpringer Journals

Published: May 17, 2017

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches

$49/month

Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588

$360/year

billed annually
Start Free Trial

14-day Free Trial