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The aim of this paper is twofold. First, we extend the results of [33]concerning the existence and uniqueness of second-order reflected 2BSDEs to thecase of two obstacles. Under some regularity assumptions on one of thebarriers, similar to the ones in [10], and when the two barriers are completelyseparated, we provide a complete wellposedness theory for doubly reflectedsecond-order BSDEs. We also show that these objects are related to non-standardoptimal stopping games, thus generalizing the connection between DRBSDEs andDynkin games first proved by Cvitanic and Karatzas [11]. More precisely, weshow under a technical assumption that the second order DRBSDEs providesolutions of what we call uncertain Dynkin games and that they also allow us toobtain super and subhedging prices for American game options (also calledIsraeli options) in financial markets with volatility uncertainty

Second-order BSDEs with general reflection and game options under uncertainty