Reflection principle and Ocone martingales

Let M = ( M t ) t ≥ 0 be any continuous real-valued stochastic process. We prove that if there exists a sequence ( a n ) n ≥ 1 of real numbers which converges to 0 and such that M satisfies the reflection property at all levels a n and 2 a n 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 a n ? We prove that this 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 ≥ 1 is trivial. We establish similar results for skip free Z -valued processes and use them for the proof in continuous time, via a discretization in space.