Characterising Ocone Local Martingales with Reflections

Let M = (M _t ) _{t ≥ 0} be any continuous real-valued stochastic process such that M ₀ = 0. Chaumont and Vostrikova proved that if there exists a sequence (a _n ) _{n ≥ 1} of positive real numbers converging to 0 such that M satisfies the reflection principle at levels 0, a _n and 2a _n , for each n ≥ 1, then M is an Ocone local martingale. They also asked whether the reflection principle at levels 0 and a _n only (for each n ≥ 1) is sufficient to ensure that M is an Ocone local martingale. We give a positive answer to this question, using a slightly different approach, which provides the following intermediate result. Let a and b be two positive real numbers such that \(a/(a + b)\) is not dyadic. If M satisfies the reflection principle at the level 0 and at the first passage-time in { − a, b}, then M is close to a local martingale in the following sense: | e[M _{S ∘ M} ] | ≤ a + b for every stopping time S in the canonical filtration of \(\mathbf{w} =\{ w \in \mathcal{C}(\mathbf{r}_{+},\mathbf{r}): w(0) = 0\}\) such that the stopped process M _{⋅ ∧ (S ∘ M)} is uniformly bounded.