Reflection principle and Ocone martingales

Let M=(Mt)t≥0M=(Mt)t≥0 be any continuous real-valued stochastic process. We prove that if there exists a sequence (an)n≥1(an)n≥1 of real numbers which converges to 0 and such that MM satisfies the reflection property at all levels anan and 2an2an with n≥1n≥1, then MM 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 anan? We prove that this question is equivalent to the fact that for Brownian motion, the σσ-field of the invariant events by all reflections at levels anan, n≥1n≥1 is trivial. We establish similar results for skip free ZZ-valued processes and use them for the proof in continuous time, via a discretization in space.