A generalized problem of optimal stopping is defined, which has a solution in the class of quasi-stopping times, which generalize the classical stopping times. The optimal quasi-stopping times are fully characterized. By time reversal, on a Markov process, various results of Meyer, Mokobodzki and Rost are interpreted as results of existence of optimal quasi-stopping times for the reversed process. Finally, a game on quasi-stopping times is solved.
"Temps D'Arret Optimal, Quasi-Temps D'Arret et Retournement du Temps." Ann. Probab. 7 (6) 933 - 964, December, 1979. https://doi.org/10.1214/aop/1176994889