Reflection principle and Ocone martingales.

Abstract : 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 $\sigma$-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.
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Submitted on : Wednesday, July 23, 2008 - 11:29:54 PM
Last modification on : Wednesday, December 19, 2018 - 2:08:04 PM
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  • HAL Id : hal-00305333, version 1
  • ARXIV : 0807.3816

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Loïc Chaumont, L. Vostrikova. Reflection principle and Ocone martingales.. 2008. 〈hal-00305333〉

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