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

• 成果类型: Article
• 署名作者: El Karoui, Mcole; Meziou, Asma
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117907000000222
• 发表日期: 2008
• 页码: 647-697
• 关键词: CONTINGENT CLAIMS; optimization; bandits

摘要

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.