We develop a martingale approach for studying continuous-time stochastic differential games of control and stopping, in a non-Markovian framework and with the control affecting only the drift term of the state-process. Under appropriate conditions, we show that the game has a value and construct a saddle pair of optimal control and stopping strategies. Crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities.
"Martingale approach to stochastic differential games of control and stopping." Ann. Probab. 36 (4) 1495 - 1527, July 2008. https://doi.org/10.1214/07-AOP367