Doubly Reflected BSDEs with default time under stochastic Lipschitz coefficients and Applications

We formulate a notion of doubly reflected BSDEs with a default time and two completely separated RCLL barriers. We demonstrate the existence and uniqueness of the solution. Within the defaultable setup, we introduce a type of generalized Dynkin game, we characterize the common value of the game via the solution of the doubly reflected BSDEs and we show the existence of a saddle point.