We establish existence and uniqueness results for adapted solutions of backward stochastic differential equations (BSDE's) with two reflecting barriers, generalizing the work of El Karoui, Kapoudjian, Pardoux, Peng and Quenez. Existence is proved first by solving a related pair of coupled optimal stopping problems, and then, under different conditions, via a penalization method. It is also shown that the solution coincides with the value of a certain Dynkin game, a stochastic game of optimal stopping. Moreover, the connection with the backward SDE enables us to provide a pathwise (deterministic) approach to the game.
"Backward stochastic differential equations with reflection and Dynkin games." Ann. Probab. 24 (4) 2024 - 2056, October 1996. https://doi.org/10.1214/aop/1041903216