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

## TL;DR

This paper introduces a nonlinear optional decomposition for $ ext{Y}^{g,\xi}$-submartingales in jump-diffusion models and applies it to characterize the superhedging price of American options from the buyer's perspective in incomplete markets.

## Contribution

It provides the first infinitesimal characterization of the buyer's superhedging price using a maximal subsolution of a constrained reflected BSDE, extending the theory of American option hedging.

## Key findings

- Established nonlinear optional decomposition for $ ext{Y}^{g,\xi}$-submartingales.
- Derived an infinitesimal characterization of the buyer's superhedging price.
- Connected the superhedging problem to maximal subsolutions of constrained reflected BSDEs.

## Abstract

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}.