| Popis: |
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. |
