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We study the superhedging problem for American options with completely irregular payoffs in a non-linear and incomplete market model with default. We give a dual representation of the seller's (superhedging) price in terms of the value of a non-linear mixed control/stopping problem, involving a suitable set of equivalent probability measures. We characterize the seller's price process as the minimal supersolution of two types of reflected BSDEs: a constrained one and an optional one. Under some regularity assumptions on the pay-off, we show a duality result for the buyer's price in terms of the value of a non-linear control/stopping game problem. [ABSTRACT FROM AUTHOR]

Abstract: We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). The financial position is given by an RCLL adapted process. We first state some properties of RBSDEs with jumps when the obstacle process is RCLL only. We then prove that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of optimal stopping times is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, we investigate robust optimal stopping problems related to the case with model ambiguity and their links with mixed control/optimal stopping game problems. We prove that, under some hypothesis, the value function is equal to the solution of an RBSDE. We then study the existence of saddle points when the obstacle is left-upper semi-continuous along stopping times. [Copyright &y& Elsevier]

Abstract: In the Brownian case, the links between dynamic risk measures and BSDEs have been widely studied. In this paper, we consider the case with jumps. We first study the properties of BSDEs driven by a Brownian motion and a Poisson random measure. In particular, we provide a comparison theorem under quite weak assumptions, extending that of Royer [21]. We then give some properties of dynamic risk measures induced by BSDEs with jumps. We provide a representation property of such dynamic risk measures in the convex case as well as some results on a robust optimization problem in the case of model ambiguity. [Copyright &y& Elsevier]

We consider the optimal stopping problem with non-linear f -expectation (induced by a BSDE) without making any regularity assumptions on the payoff process ξ and in the case of a general filtration. We show that the value family can be aggregated by an optional process Y. We characterize the process Y as the E f -Snell envelope of ξ. We also establish an infinitesimal characterization of the value process Y in terms of a Reflected BSDE with ξ as the obstacle. To do this, we first establish some useful properties of irregular RBSDEs, in particular an existence and uniqueness result and a comparison theorem. [ABSTRACT FROM AUTHOR]