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Martingales and Super-Martingales Relative to a Convex Set of Equivalent Measures

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Document pages: 35 pages

Abstract: In the paper, the martingales and super-martingalesrelative to a convex set of equivalent measures are systematically studied. The notion of local regular super-martingalerelative to a convex set of equivalent measures is introduced and the necessaryand sufficient conditions of the local regularity of it in the discrete caseare founded. The description of all local regular super-martingalesrelative to a convex set of equivalent measures is presented. The notion of thecomplete set of equivalent measures is introduced. We prove that every boundedin some sense super-martingale relative to the complete set of equivalentmeasures is local regular. A new definition of the fair price of contingentclaim in an incomplete market is given and the formula for the fair price ofStandard Option of European type is found. The proved Theorems are thegeneralization of the famous Doob decomposition for super-martingale onto thecase of super-martingales relative to a convex set of equivalent measures.

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