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Decomposition Max-Plus des surmartingales et ordre convexe. Application aux options Americaines et a l'assurance de portefeuille.

Abstract : We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any quasi-left-continuous 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. In fact, using the Max-Plus supermartingale decomposition, we suggest a new approach to the classic utility maximization problem with American constraints. To do so, we transform the problem into a constrained martingale one, whose aim is to dominate an obstacle, or equivalently its Snell envelope on every intermediate date. The optimization is performed with respect to the stochastic convex order on the terminal value, which avoids any arbitrary assumption regarding the form of the agent's utility function. The "Max-Plus martingale" is shown to be optimal and this is illustrated by an explicit example based on the geometric Brownian motion. Furthermore, we exploit the links between the Azéma-Yor martingales and the Max-Plus decomposition to solve some portfolio optimization problems with state constraints and some ones related to perpetual American options. In particular, most of the classic results concerning the American boundaries of Lévy processes are shown in an elementary way. The last chapter is devoted to the pricing of Swing options, using new numerical methods.
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Asma Meziou. Decomposition Max-Plus des surmartingales et ordre convexe. Application aux options Americaines et a l'assurance de portefeuille.. Mathématiques [math]. Ecole Polytechnique X, 2006. Français. ⟨pastel-00002177⟩

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