  We study a doubly reflected backward stochastic differential equation (BSDE)
with integrable parameters and the related Dynkin game. When the lower obstacle
$L$ and the upper obstacle $U$ of the equation are completely separated, we
construct a unique solution of the doubly reflected BSDE by pasting local
solutions and show that the $Y-$component of the unique solution represents the
value process of the corresponding Dynkin game under $g-$evaluation, a
nonlinear expectation induced by BSDEs with the same generator $g$ as the
doubly reflected BSDE concerned. In particular, the first time when process $Y
$ meets $L$ and the first time when process $Y $ meets $U$ form a saddle point
of the Dynkin game.
