Dynamic Games and Applications

, Volume 8, Issue 2, pp 280–314 | Cite as

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



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.


Zero-sum continuous-time game Incomplete information Hamilton–Jacobi equations Brownian motion Measure-valued process 

Mathematics Subject Classification

91A05 91A23 49N70 



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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Manufacture des Tabacs Allée de BrienneToulouse Cedex 6France
  2. 2.Université de Bretagne OccidentaleBrest CedexFrance

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