In this paper we study the problem of nonlinear pricing of an American option with a right-continuous left-limited (RCLL) payoff process in an incomplete market with default, from the buyer's point of view. We show that the buyer's price process can be represented as the value of a stochastic control/optimal stopping game problem with nonlinear expectations, which corresponds to the maximal subsolution of a constrained reflected Backward Stochastic Differential Equation (BSDE). We then deduce a nonlinear optional decomposition of the buyer's price process. To the best of our knowledge, no dynamic dual representation (resp. no optional decomposition) of the buyer's price process can be found in the literature, even in the case of a linear incomplete market and brownian filtration. Finally, we prove the "infimum" and the "supremum" in the definition of the stochastic game problem can be interchanged. Our method relies on new tools, as simultaneous nonlinear Doob-Meyer decompositions of processes which have a $\mathscr{Y}^\nu$-submartingale property for each admissible control $\nu$.
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