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 LL and the upper obstacle UU of the equation are completely separated, we construct a unique solution of the doubly reflected BSDE by pasting local solutions, and show that the YY-component of the unique solution represents the value process of the corresponding Dynkin game under gg-evaluation, a nonlinear expectation induced by BSDEs with the same generator gg as the doubly reflected BSDE concerned. In particular, the first time τ∗τ∗ when process YY meets LL and the first time γ∗γ∗ when process YY meets UU form a saddle point of the Dynkin game.