Games with Incomplete Information in Continuous Time and for Continuous Types

We consider a two-player zero-sum game with integral payoff and with incomplete information on one side, where the payoff is chosen among a continuous set of possible payoffs. We prove that the value function of this game is solution of an auxiliary optimization problem over a set of measure-valued processes. Then we use this equivalent formulation to characterize the value function as the viscosity solution of a special type of a Hamilton-Jacobi equation. This paper generalizes the results of a previous work of the authors [5], where only a finite number of possible payoffs is considered. Key-words : Continuous-time game, incomplete information, martingale measures, Hamilton-Jacobi equations. A.M.S. classification : 91A05, 91A23, 60G60