  In this paper, we focus on the stochastic representation of a system of
coupled Hamilton-Jacobi-Bellman-Isaacs (HJB-Isaacs (HJBI), for short) equations
which is in fact a system of coupled Isaacs' type integral-partial differential
equation. For this, we introduce an associated zero-sum stochastic differential
game, where the state process is described by a classical stochastic
differential equation (SDE, for short) with jumps, and the cost functional of
recursive type is defined by a new type of backward stochastic differential
equation (BSDE, for short) with two Poisson random measures, whose
wellposedness and a prior estimate as well as the comparison theorem are
investigated for the first time. One of the Poisson random measures appearing
in the SDE and the BSDE stems from the integral term of the HJBI equations; the
other random measure in BSDE is introduced to link the coupling factor of the
HJBI equations. We show through an extension of the dynamic programming
principle that the lower value function of this game problem is the viscosity
solution of the system of our coupled HJBI equations. The uniqueness of the
viscosity solution is also obtained in a space of continuous functions
satisfying certain growth condition. In addition, also the upper value function
of the game is shown to be the solution of the associated system of coupled
Issacs' type of integral-partial differential equations. As a byproduct, we
obtain the existence of the value for the game problem under the well-known
Isaacs' condition.
