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In the paper, we introduce the notion of a local regular supermartingalerelative to a convex set of equivalent measures and prove for it the necessaryand sufficient conditions of optional Doob decomposition in the discrete case.This Theorem is a generalization of the famous Doob decomposition onto the caseof supermartingales relative to a convex set of equivalent measures. Thedescription of all local regular supermartingales relative to a convex set ofequivalent measures is presented. A notion of complete set of equivalentmeasures is introduced. We prove that every non negative boundedsupermartingale relative to a complete set of equivalent measures is localregular. A new definition of fair price of contingent claim in incompletemarket is given and a formula for fair price of Standard option of Europeantype is found.

Generalization of Doob Decomposition Theorem and Risk Assessment in Incomplete Markets