Before revision:  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 .
After revision:  Under appropriate conditions, we show that the game has a value and the value function is the unique viscosity solution to an obstacle problem for a Hamilton-Jacobi-Bellman equation , by generalizing the weak dynamic programming principles in 3 .