A zero-sum hybrid stochastic differential game with impulse controls

In this paper, we study a zero-sum stochastic differential game with the following salient features: (i) the system state is dictated by a hybrid diffusion, (ii) both players use impulse controls, and (iii) the game takes place on an infinite time horizon. First, the dynamic programming principle for the problem is proven. Then, the lower and upper value functions of the game are characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, which turns out to be a coupled system of variational inequalities with bilateral obstacles. Moreover, a verification theorem as a sufficient condition to identify a Nash equilibrium is established. The Nash equilibrium strategies for the two players, indicating when and how it is optimal to intervene, are given in terms of the obstacle part of the HJBI equation.