  We analyze the continuous time zero-sum and cooperative controller-stopper
games of Karatzas and Sudderth [Annals of Probability, 2001], Karatzas and
Zamfirescu [Annals of Probability, 2008] and Karatzas and Zamfirescu [Applied
Mathematics and Optimization, 2005] when the volatility of the state process is
controlled as in Bayraktar and Huang [SIAM Journal on Control and Optimization,
2013] but additionally when the state process has controlled jumps. We perform
this analysis by first resolving the stochastic target problems (of Soner and
Touzi [SIAM Journal on Control and Optimization, 2002; Journal of European
Mathematical Society, 2002]) with a cooperative or a non-cooperative stopper
and then embedding the original problem into the latter set-up. Unlike in
Bayraktar and Huang [SIAM Journal on Control and Optimization, 2013] our
analysis relies crucially on the Stochastic Perron method of Bayraktar and
Sirbu [SIAM Journal on Control and Optimization, 2013] but not the dynamic
programming principle, which is difficult to prove directly for games.
