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Doubly Reflected BSDEs with Integrable Parameters and Related Dynkin Games

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.