  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 [8]), and our
result is a continuous-time analog of the so called splitting-game introduced
in Laraki [20] and Sorin [27] in order to analyze discrete-time models. It was
proved by Cardaliaguet [4, 5] 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.
