  We consider a stochastic differential game in the context of forward-backward
stochastic differential equations, where one player implements an impulse
control while the opponent controls the system continuously. Utilizing the
notion of "backward semigroups" we first prove the dynamic programming
principle (DPP) for a truncated version of the problem in a straightforward
manner. Relying on a uniform convergence argument then enables us to show the
DPP for the general setting. In particular, this avoids technical constraints
imposed in previous works dealing with the same problem. Moreover, our approach
allows us to consider impulse costs that depend on the present value of the
state process in addition to unbounded coefficients.
  Using the dynamic programming principle we deduce that the upper and lower
value functions are both solutions (in viscosity sense) to the same
Hamilton-Jacobi-Bellman-Isaacs obstacle problem. By showing uniqueness of
solutions to this partial differential inequality we conclude that the game has
a value.

Optimization and Control