A Differential Game with a Blind Player

Abstract

We consider a zero sum differential game with lack of observation on one side. The initial state of the system is drawn at random according to a probability $\mu_0$ on ${\mathbb R}^N$. Player I is informed of the initial position of state, while Player II knows only $\mu_0$. Moreover Player I observes Player II's moves, while Player II is blind and has no further information. We prove that in this game with a terminal payoff the value exists and is characterized as the unique viscosity solution of a Hamilton--Jacobi equation on a space of probability measures.