International audienceWe introduce a game problem which can be seen as a generalization of the classical Dynkin game problemto the case of a nonlinear expectation ${\cal E}^g$, induced by a Backward Stochastic Differential Equation (BSDE) with jumps with nonlinear driver $g$. Let $\xi, \zeta$ be two RCLL adapted processes with $\xi \leq \zeta$. The criterium is given by $ {\cal J}_{\tau, \sigma}= {\cal E}^g_{0, \tau \wedge \sigma } \left(\xi_{\tau}\textbf{1}_{\{ \tau \leq \sigma\}}+\zeta_{\sigma}\textbf{1}_{\{\sigma<\tau\}}\right)$ where $\tau$ and $ \sigma$ are stopping times valued in $[0,T]$. Under Mokobodzki's condition, we establish the existence of a value function for this game, i.e. $\inf_{\sigma}\sup_{\tau} {\cal J}_{\tau, \sigma} = \sup_{\tau} \inf_{\sigma} {\cal J}_{\tau, \sigma}$. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates.When $\xi$ and $\zeta$ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping.We then study the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles

We study the properties of nonlinear Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale measure associated with a default jump with intensity process (λ t ). We give a priori estimates for these equations and prove comparison and strict comparison theorems. These results are generalized to drivers involving a singular process. The special case of a λ-linear driver is studied, leading to a representation of the solution of the associated BSDE in terms of a conditional expectation and an adjoint exponential semi-martingale. We then apply these results to nonlinear pricing of European contingent claims in an imperfect financial market with a totally defaultable risky asset. The case of claims paying dividends is also studied via a singular process.

We study pricing and superhedging strategies for game options in an imperfect market with default. We extend the results obtained by Kifer in [23] 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 [14], expressed with a nonlinear expectation induced by a nonlinear BSDE 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 super-hedged, whatever the model is.

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We study pricing and (super)hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process (ξ t ). We define the seller's superhedging price of the American option as the minimum of the initial capitals which allow the seller to build up a superhedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with nonlinear expectations induced by BSDEs with default jump, which corresponds to the solution of a reflected BSDE with lower barrier. Moreover, we show the existence of a superhedging portfolio strategy. We then consider the buyer's superhedging price, which is defined as the supremum of the initial wealths which allow the buyer to select an exercise time τ and a portfolio strategy ϕ so that he/she is superhedged. Under the additional assumption of left upper semicontinuity along stopping times of (ξ t ), we show the existence of a superhedge (τ, ϕ) for the buyer, as well as a characterization of the buyer's superhedging price via the solution of a nonlinear reflected BSDE with upper barrier.