Differential games with asymmetric information and without Isaacs' condition

We investigate a two-player zero-sum differential game with asymmetric information on the payoff and without Isaacs' condition. The dynamics is an ordinary differential equation parametrized by two controls chosen by the players. Each player has a private information on the payoff of the game, while his opponent knows only the probability distribution on the information of the other player. We show that a suitable definition of random strategies allows to prove the existence of a value in mixed strategies. This value is taken in the sense of the limit of any time discretization, as the mesh of the time partition tends to zero. We characterize it in terms of the unique viscosity solution in some dual sense of a Hamilton-Jacobi-Isaacs equation. Here we do not suppose the Isaacs' condition, which is usually assumed in differential games.