A Probabilistic Representation for the Value of Zero-Sum Differential Games with Incomplete Information on Both Sides

We prove that for a class of zero-sum differential games with incomplete information on both sides, the value admits a probabilistic representation as the value of a zero-sum stochastic differential game with complete information, where both players control a continuous martingale. A similar representation as a control problem over discontinuous martingales was known for games with incomplete information on one side (see Cardaliaguet-Rainer [9]), and our result is a continuous-time analog of the so called splitting-game introduced in Laraki [22] and Sorin [29] in order to analyze discrete-time models. It was proved by Cardaliaguet [5, 6] that the value of the games we consider is the unique solution of some Hamilton-Jacobi equation with convexity constraints. Our result provides therefore a new probabilistic representation for solutions of Hamilton-Jacobi equations with convexity constraints as values of stochastic differential games with unbounded control spaces and unbounded volatility. Key-words : Zero-sum continuous-time game, incomplete information, HamiltonJacobi equations, stochastic differential game A.M.S. classification : 91A05, 91A23, 60G60