Existence of value for differential games with incomplete information and signals on initial states and payoffs

This article is devoted to proving the existence of the value for a two-players differential game with incomplete information and a signal structure on the data (initial positions, dynamics, and payoffs). Before the game starts, the data are chosen randomly in a finite set. The players are not informed on the exact data but they receive a signal. The main result of the paper says that the value of the game does exist for suitably defined nonanticipative random strategies. To obtain such a result, we introduce and study a new type of second-order double-obstacle Hamilton–Jacobi–Isaacs equation. We also prove that the value is the unique viscosity solution of this partial differential equation.