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A Two-Player Zero-sum Game Where Only One Player Observes a Brownian Motion

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Abstract

We study a two-player zero-sum game in continuous time, where the payoff—a running cost—depends on a Brownian motion. This Brownian motion is observed in real time by one of the players. The other one observes only the actions of his/her opponent. We prove that the game has a value and characterize it as the largest convex subsolution of a Hamilton–Jacobi equation on the space of probability measures.

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Acknowledgements

We thank Pierre Cardaliaguet for very fruitful discussions. We thank the referees for their careful reading and their pertinent remarks. This work has been partially supported by the French National Research Agency ANR-16-CE40-0015-01 MFG.

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Authors and Affiliations

  1. Manufacture des Tabacs Allée de Brienne, 31015, Toulouse Cedex 6, France

    Fabien Gensbittel

  2. Université de Bretagne Occidentale, 6, avenue Victor-le-Gorgeu, B.P. 809, 29285, Brest Cedex, France

    Catherine Rainer

Authors
  1. Fabien Gensbittel
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  2. Catherine Rainer
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Correspondence to Catherine Rainer.

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Gensbittel, F., Rainer, C. A Two-Player Zero-sum Game Where Only One Player Observes a Brownian Motion. Dyn Games Appl 8, 280–314 (2018). https://doi.org/10.1007/s13235-017-0219-5

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  • DOI: https://doi.org/10.1007/s13235-017-0219-5

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