Reflection principle and Ocone martingales

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