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.