Reflection principle and Ocone martingales
| dc.creator | Chaumont, Loïc | |
| dc.creator | Vostrikova, L. | |
| dc.date | 2008-07-24 | |
| dc.date.accessioned | 2026-07-07T09:52:34Z | |
| dc.date.available | 2026-07-07T09:52:34Z | |
| dc.description | Let $M =(M_t)_{t\geq 0}$ be any continuous real-valued stochastic process. We prove that if there exists a sequence $(a_n)_{n\geq 1}$ of real numbers which converges to 0 and such that $M$ satisfies the reflection property at all levels $a_n$ and $2a_n$ with $n\geq 1$, then $M$ is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels $a_n$? Then we prove that the later question is equivalent to the fact that for Brownian motion, the $σ$-field of the invariant events by all reflections at levels $a_n$, $n\ge1$ is trivial. We establish similar results for skip free $\mathbb{Z}$-valued processes and use them for the proof in continuous time, via a discretisation in space. | |
| dc.identifier | https://arxiv.org/abs/0807.3816 | |
| dc.identifier | http://arxiv.org/abs/0807.3816 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/165639 | |
| dc.subject | Probability | |
| dc.subject | 60G44, 60G42, 60J65 | |
| dc.title | Reflection principle and Ocone martingales | |
| dc.type | text |