Stochastic representation for solutions of a system of coupled HJB-Isaacs equations with integral–differential operators

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 $\mu$ 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 Isaacs’ 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.