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March 2008 Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance
Nicole El Karoui, Asma Meziou
Ann. Probab. 36(2): 647-697 (March 2008). DOI: 10.1214/009117907000000222

Abstract

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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Nicole El Karoui. Asma Meziou. "Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance." Ann. Probab. 36 (2) 647 - 697, March 2008. https://doi.org/10.1214/009117907000000222

Information

Published: March 2008
First available in Project Euclid: 29 February 2008

zbMATH: 1152.60038
MathSciNet: MR2393993
Digital Object Identifier: 10.1214/009117907000000222

Subjects:
Primary: 16Y60, 60E15, 60G07, 60G40, 60G51
Secondary: 60G44, 91B28

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 2 • March 2008
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