Annals of Probability

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

Nicole El Karoui and Asma Meziou

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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.

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Ann. Probab., Volume 36, Number 2 (2008), 647-697.

First available in Project Euclid: 29 February 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G07: General theory of processes 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G51: Processes with independent increments; Lévy processes 16Y60: Semirings [See also 12K10] 60E15: Inequalities; stochastic orderings
Secondary: 91B28 60G44: Martingales with continuous parameter

Supermartingale decompositions Max-Plus algebra running supremum process American options optimal stopping Lévy processes convex order martingale optimization with constraints portfolio insurance Azéma–Yor martingales


El Karoui, Nicole; Meziou, Asma. Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance. Ann. Probab. 36 (2008), no. 2, 647--697. doi:10.1214/009117907000000222.

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