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We are concerned with a new type of supermartingale decomposition in theMax-Plus algebra, which essentially consists in expressing any supermartingaleof class $(\mathcal{D})$ as a conditional expectation of some running supremumprocess. As an application, we show how the Max-Plus supermartingaledecomposition allows, in particular, to solve the American optimal stoppingproblem without having to compute the option price. Some illustrative examplesbased on one-dimensional diffusion processes are then provided. Anotherinteresting application concerns the portfolio insurance. Hence, based on the``Max-Plus martingale,'' we solve in the paper an optimization problem whoseaim is to find the best martingale dominating a given floor process (on everyintermediate date), w.r.t. the convex order on terminal values.

Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance