Random walks and Lévy processes as rough paths. (English) Zbl 1436.60087
‘The paper focuses on several questions regarding Lévy processes and random walks in homogeneous groups, with a particular focus on apllications to rough path theory.’ The author considers ‘random walks and Lévy processes in a homogeneous group \(G\). For all \(p>0\), (he) completely characterise (almost) all \(G\)-valued Lévy processes whose sample paths have finite \(p\)-variation, and give(s) 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 \(\mathbb R^d\), so that processes of finite \(p\)-variation are identified with rough paths, (he) demonstrate(s) applications of (his) results to weak convergence of stochastic flows and provide(s) a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process. At the heart of (his) analysis is a criterion for tightness of \(p\)-variation for a collection of càdlàg strong Markov processes’. In contrast to earlier contributions on the subject, the proofs rely on the approximation of the Lévy process by a sequence of random walks. A crucial result for the analysis generalizes a theorem of M. Manstavicius on strong \(p\)-variation of Markov processes.
Reviewer: Alexander Schnurr (Siegen)
MSC:
| 60L20 | Rough paths |
| 60G51 | Processes with independent increments; Lévy processes |
| 60L50 | Rough partial differential equations |
Keywords:
homogeneous groups; rough paths; Lévy processes; random walks; tightness of \(p\)-variation; stochastic flows; characteristic functions of signaturesReferences:
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