Superhedging prices of European and American options in a non-linear incomplete market with default

This paper studies the superhedging prices and the associated superhedging strategies for European and American options in a non-linear incomplete market with default. We present the seller's and the buyer's point of view. The underlying market model consists of a risk-free asset and a risky asset driven by a Brownian motion and a compensated default martingale. The portfolio process follows non-linear dynamics with a non-linear driver ƒ. By using a dynamic programming approach, we first provide a dual formulation of the seller's (superhedging) price for the European option as the supremum over a suitable set of equivalent probability measures *Q* ∈ $\mathcal{Q}$ of the ƒ-evaluation/expectation under *Q* of the payoff. We also provide an infinitesimal characterization of this price as the minimal supersolution of a constrained BSDE with default. By a form of symmetry, we derive corresponding results for the buyer. We also give a dual representation of the seller's (superhedging) price for the American option associated with an irregular payoff (ξ *t* ) (not necessarily càdlàg) in terms of the value of a non-linear mixed control/stopping problem. We also provide an infinitesimal characterization of this price in terms of a constrained reflected BSDE. When ξ is càdlàg, we show a duality result for the buyer's price. These results rely on first establishing a non-linear optional decomposition for processes which are $\mathcal{E}$ ƒ -strong supermartingales under *Q*, for all *Q* ∈ $\mathcal{Q}$ .