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

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Publisher
Springer Journals
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

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