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Generalization of Doob Decomposition Theorem and Risk Assessment in Incomplete Markets

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  • N. S. Gonchar

Abstract

In the paper, we introduce the notion of a local regular supermartingale relative to a convex set of equivalent measures and prove for it the necessary and sufficient conditions of optional Doob decomposition in the discrete case. This Theorem is a generalization of the famous Doob decomposition onto the case of supermartingales relative to a convex set of equivalent measures. The description of all local regular supermartingales relative to a convex set of equivalent measures is presented. A notion of complete set of equivalent measures is introduced. We prove that every non negative bounded supermartingale relative to a complete set of equivalent measures is local regular. A new definition of fair price of contingent claim in incomplete market is given and a formula for fair price of Standard option of European type is found.

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  • N. S. Gonchar, 2016. "Generalization of Doob Decomposition Theorem and Risk Assessment in Incomplete Markets," Papers 1611.09062, arXiv.org.
    • Handle: RePEc:arx:papers:1611.09062
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    References listed on IDEAS

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    4. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
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