We consider a zero-sum stochastic differential controller-and-stopper game in
which the state process is a controlled jump-diffusion evolving in a
multi-dimensional Euclidean space. In this game, the controller affects both
the drift and the volatility terms of the state process. Under appropriate
conditions, we show that the lower value function of this game is a viscosity
solution to an obstacle problem for a Hamilton-Jacobi-Bellman equation, by
generalizing the weak dynamic programming principles in [3].