  We study a zero-sum stochastic differential game (SDG) in which one
controller plays an impulse control while their opponent plays a stochastic
control. We consider an asymmetric setting in which the impulse player commits
to, at the start of the game, performing less than q impulses (q can be chosen
arbitrarily large). In order to obtain the uniform continuity of the value
functions, previous works involving SDGs with impulses assume the cost of an
impulse to be decreasing in time. Our work avoids such restrictions by
requiring impulses to occur at rational times. We establish that the resulting
game admits a value, and in turn, the existence and uniqueness of viscosity
solutions to an associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational
inequality.
