  In this paper we study zero-sum two-player stochastic differential games with
the help of theory of Backward Stochastic Differential Equations (BSDEs). At
the one hand we generalize the results of the pioneer work of Fleming and
Souganidis by considering cost functionals defined by controlled BSDEs and by
allowing the admissible control processes to depend on events occurring before
the beginning of the game (which implies that the cost functionals become
random variables), on the other hand the application of BSDE methods, in
particular that of the notion of stochastic "backward semigroups" introduced by
Peng allows to prove a dynamic programming principle for the upper and the
lower value functions of the game in a straight-forward way, without passing by
additional approximations. The upper and the lower value functions are proved
to be the unique viscosity solutions of the upper and the lower
Hamilton-Jacobi-Bellman-Isaacs equations, respectively. For this Peng's BSDE
method is translated from the framework of stochastic control theory into that
of stochastic differential games.
