Random walks and Levy processes as rough paths

• 成果类型: Article
• 署名作者: Chevyrev, Ilya
• 署名单位: University of Oxford
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0781-1
• 发表日期: 2018
• 页码: 891-932
• 关键词: independent increments; MARKOV-PROCESSES; uniqueness; signature

摘要

We consider random walks and Levy processes in a homogeneous group G. For all p > 0, we completely characterise (almost) all G-valued Levy 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 Levy process in p-variation topology. In the case that G is the free nilpotent Lie group over 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 Levy-Khintchine formula for the characteristic function of the signature of a Levy process. At the heart of our analysis is a criterion for tightness of p-variation for a collection of cadlag strong Markov processes.