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Reflection principle and Ocone martingales

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  • Chaumont, L.
  • Vostrikova, L.

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 [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.

Suggested Citation

  • Chaumont, L. & Vostrikova, L., 2009. "Reflection principle and Ocone martingales," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3816-3833, October.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:10:p:3816-3833
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    Cited by:

    1. Zhang, Jichen & Chen, Zengjing, 2023. "A transitivity property of Ocone martingales," Statistics & Probability Letters, Elsevier, vol. 193(C).

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