We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows, in particular, to solve the American optimal stopping problem without having to compute the option price. Some illustrative examples based on one-dimensional diffusion processes are then provided. Another interesting application concerns the portfolio insurance. Hence, based on the “Max-Plus martingale,” we solve in the paper an optimization problem whose aim is to find the best martingale dominating a given floor process (on every intermediate date), w.r.t. the convex order on terminal values.
@article{1204306963,
author = {El Karoui, Nicole and Meziou, Asma},
title = {Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance},
journal = {Ann. Probab.},
volume = {36},
number = {1},
year = {2008},
pages = { 647-697},
language = {en},
url = {http://dml.mathdoc.fr/item/1204306963}
}
El Karoui, Nicole; Meziou, Asma. Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance. Ann. Probab., Tome 36 (2008) no. 1, pp. 647-697. http://gdmltest.u-ga.fr/item/1204306963/