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We formulate a notion of doubly reflected BSDE in the case where the barriers$\xi$ and $\zeta$ do not satisfy any regularity assumption and with a generalfiltration. Under a technical assumption (a Mokobodzki-type condition), we showexistence and uniqueness of the solution. In the case where $\xi$ is rightupper-semicontinuous and $\zeta$ is right lower-semicontinuous, the solution ischaracterized in terms of the value of a corresponding $\mathcal{E}^f$-Dynkingame, i.e. a game problem over stopping times with (non-linear)$f$-expectation, where $f$ is the driver of the doubly reflected BSDE. In thegeneral case where the barriers do not satisfy any regularity assumptions, thesolution of the doubly reflected BSDE is related to the value of ''anextension'' of the previous non-linear game problem over a larger set of''stopping strategies'' than the set of stopping times. This characterizationis then used to establish a comparison result and \textit{a priori} estimateswith universal constants.

Doubly Reflected BSDEs and ${\cal E}^{f}$-Dynkin games: beyond the right-continuous case