  In the present paper, we study a two-player zero-sum deterministic
differential game with both players adopting impulse controls, in infinite time
horizon, under rather weak assumptions on the cost functions. We prove by means
of the dynamic programming principle (DPP) that the lower and upper value
functions are continuous and viscosity solutions to the corresponding
Hamilton-Jacobi-Bellman-Isaacs (HJBI) quasi-variational inequality (QVI). We
define a new HJBI QVI for which, under a proportional property assumption on
the maximizer cost, the value functions are the unique viscosity solution. We
then prove that the lower and upper value functions coincide.
