Game options in an imperfect market with default

We study pricing and superhedging strategies for game options in an imperfect market with default. We extend the results obtained by Kifer in [Game Options, Finance. Stoch., 4 (2000), pp. 443-463] in the case of a perfect market model to the case of an imperfect market with default, when the imperfections are taken into account via the nonlinearity of the wealth dynamics. We introduce the seller's price of the game option as the infimum of the initial wealths which allow the seller to be superhedged. We prove that this price coincides with the value function of an associated generalized Dynkin game, recently introduced in [R. Dumitrescu, M.-C. Quenez, and A. Sulem, Elect. J. Probab., 21 (2016), 64], expressed with a nonlinear expectation induced by a nonlinear backward SDE with default jump. We, moreover, study the existence of superhedging strategies. We then address the case of ambiguity on the model-for example ambiguity on the default probability-and characterize the robust seller's price of a game option as the value function of a mixed generalized Dynkin game. We study the existence of a cancellation time and a trading strategy which allow the seller to be superhedged, whatever the model is.