  In this paper, we study a class of zero-sum two-player stochastic
differential games with the controlled stochastic differential equations and
the payoff/cost functionals of recursive type. As opposed to the pioneering
work by Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989),
pp.~293--314] and the seminal work by Buckdahn and Li [SIAM J. Control Optim.,
47 (2008), pp.~444--475], the involved coefficients may be random, going beyond
the Markovian framework and leading to the random upper and lower value
functions. We first prove the dynamic programming principle for the game, and
then under the standard Lipschitz continuity assumptions on the coefficients,
the upper and lower value functions are shown to be the viscosity solutions of
the upper and the lower fully nonlinear stochastic
Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. A stability
property of viscosity solutions is also proved. Under certain additional
regularity assumptions on the diffusion coefficient, the uniqueness of the
viscosity solution is addressed as well.
