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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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Loïc Chaumont, L. Vostrikova. Reflection principle and Ocone martingales.. Stochastic Processes and their Applications, Elsevier, 2009, 119 (10), pp.3816-3833. ⟨10.1016/j.spa.2009.07.009⟩. ⟨hal-00305333v2⟩

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