An optional decomposition of $\mathscr{Y}^{g,\xi}-submartingales$ and applications to the hedging of American options in incomplete markets
In the recent paper [13], the notion of Y g,ξ -submartingale processes has been introduced. Within a jump-diﬀusion model, we prove here that a process X which satisﬁes the simultaneous Y Q ,g,ξ submartingale property under a suitable family of equivalent probability measures Q , admits a nonlinear optional decomposition . This is an analogous result to the well known optional decomposition of simultaneous (classical and 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 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 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 characterization in terms of the maximal subsolution of a constrained reﬂected BSDE .