A two player zerosum game where only one player observes a Brownian motion

TL;DR

This paper analyzes a continuous-time two-player zero-sum game involving a Brownian motion observed by only one player, establishing the existence of a game value and characterizing it via a Hamilton-Jacobi equation on probability measures.

Contribution

It introduces a novel game model with asymmetric information and provides a mathematical characterization of its value through advanced PDE techniques.

Findings

The game has a well-defined value.

The value is characterized as the largest convex subsolution of a Hamilton-Jacobi equation.

The model advances understanding of asymmetric information in stochastic games.

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 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.