We are concerned with a new type of supermartingale decomposition in the
Max-Plus algebra, which essentially consists in expressing any supermartingale
of class (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.Comment: Published in at http://dx.doi.org/10.1214/009117907000000222 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org