  This paper is concerned with a new type of differential game problems of
forwardbackward stochastic systems. There are three distinguishing features:
Firstly, our game systems are forward-backward doubly stochastic differential
equations, which is a class of more general game systems than other
forward-backward stochastic game systems without doubly stochastic terms;
Secondly, forward equations are directly related to backward equations at
initial time, not terminal time; Thirdly, the admissible control is required to
be adapted to a sub-information of the full information generated by the
underlying Brownian motions. We give a necessary and a sufficient conditions
for both an equilibrium point of nonzero-sum games and a saddle point of
zero-sum games. Finally, we work out an example of linear-quadratic nonzero-sum
differential games to illustrate the theoretical applications. Applying some
stochastic filtering techniques, we obtain the explicit expression of the
equilibrium point.
