Random walks and Lévy processes as rough paths
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 L...
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| Materyal Türü: | Journal article |
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Springer Verlag
2017
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| _version_ | 1826268675076784128 |
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| author | Chevyrev, I |
| author_facet | Chevyrev, I |
| author_sort | Chevyrev, I |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T21:13:19Z |
| format | Journal article |
| id | oxford-uuid:3eebfa48-d96e-428c-907b-b83c15535138 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T21:13:19Z |
| publishDate | 2017 |
| publisher | Springer Verlag |
| record_format | dspace |
| spelling | oxford-uuid:3eebfa48-d96e-428c-907b-b83c155351382022-03-26T14:28:40ZRandom walks and Lévy processes as rough pathsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3eebfa48-d96e-428c-907b-b83c15535138Symplectic Elements at OxfordSpringer Verlag2017Chevyrev, IWe 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. |
| spellingShingle | Chevyrev, I Random walks and Lévy processes as rough paths |
| title | Random walks and Lévy processes as rough paths |
| title_full | Random walks and Lévy processes as rough paths |
| title_fullStr | Random walks and Lévy processes as rough paths |
| title_full_unstemmed | Random walks and Lévy processes as rough paths |
| title_short | Random walks and Lévy processes as rough paths |
| title_sort | random walks and levy processes as rough paths |
| work_keys_str_mv | AT chevyrevi randomwalksandlevyprocessesasroughpaths |