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

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  • Nicole El Karoui
  • Asma Meziou
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    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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    File URL: http://arxiv.org/pdf/0804.2561
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0804.2561.

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    Date of creation: Apr 2008
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    Publication status: Published in Annals of Probability 2008, Vol. 36, No. 2, 647-697
    Handle: RePEc:arx:papers:0804.2561

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    Web page: http://arxiv.org/

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    1. Kramkov, D.O., 1994. "Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets," Discussion Paper Serie B, University of Bonn, Germany 294, University of Bonn, Germany.
    2. El Karoui, Nicole & Jeanblanc, Monique & Lacoste, Vincent, 2005. "Optimal portfolio management with American capital guarantee," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 29(3), pages 449-468, March.
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    Cited by:
    1. Christensen, Sören & Salminen, Paavo & Ta, Bao Quoc, 2013. "Optimal stopping of strong Markov processes," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 123(3), pages 1138-1159.
    2. Marina Di Giacinto & Salvatore Federico & Fausto Gozzi, 2011. "Pension funds with a minimum guarantee: a stochastic control approach," Finance and Stochastics, Springer, Springer, vol. 15(2), pages 297-342, June.

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