  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 general
filtration. Under a technical assumption (a Mokobodzki-type condition), we show
existence and uniqueness of the solution. In the case where $\xi$ is right
upper-semicontinuous and $\zeta$ is right lower-semicontinuous, the solution is
characterized in terms of the value of a corresponding $\mathcal{E}^f$-Dynkin
game, i.e. a game problem over stopping times with (non-linear)
$f$-expectation, where $f$ is the driver of the doubly reflected BSDE. In the
general case where the barriers do not satisfy any regularity assumptions, the
solution of the doubly reflected BSDE is related to the value of ''an
extension'' of the previous non-linear game problem over a larger set of
''stopping strategies'' than the set of stopping times. This characterization
is then used to establish a comparison result and \textit{a priori} estimates
with universal constants.
