Reflection principle and Ocone martingales |
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Authors: | L Chaumont L Vostrikova |
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Institution: | LAREMA, Département de Mathématiques, Université d’Angers, 2, Bd Lavoisier - 49045, Angers Cedex 01, France |
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Abstract: | 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 σ-field of the invariant events by all reflections at levels an, 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. |
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Keywords: | 60G44 60G42 60J65 |
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