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An optional decomposition of $\mathscr{Y}^{g,ξ}-submartingales$ and applications to the hedging of American options in incomplete markets

In the recent paper \cite{DESZ}, the notion of $\mathscr{Y}^{g,\xi}$-submartingale processes has been introduced. Within a jump-diffusion model, we prove here that a process $X$ which satisfies the simultaneous $\mathscr{Y}^{\mathbb{Q},g,\xi}$ -submartingale property under a suitable family of equivalent probability measures $\mathbb{Q}$, admits a \textit{nonlinear optional decomposition}. This is an analogous result to the well known optional decomposition of simultaneous (classical and $\mathscr{E}^g$-)supermartingales. We then apply this decomposition to the super-hedging problem of an American option in a jump-diffusion model, from the buyer's point of view. We obtain an \textit{infinitesimal characterization} of the buyer's superhedging price, this result being completely new in the literature. Indeed, it is well known that the seller's superheding price of an American option admits an infinitesimal representation in terms of the \textit{minimal supersolution of a constrained reflected BSDE}. To the best of our knowledge, no analogous result has been established for the buyer of the American option in an incomplete market. Our results fill this gap, and show that the buyer's super-hedging price admits an infinitesimal charcaterization in terms of the \textit{maximal subsolution of a constrained reflected BSDE}.

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