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

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  • Nicholas S. Gonchar

Abstract

In the paper, the martingales and super-martingales relative to a convex set of equivalent measures are systematically studied. The notion of local regular super-martingale relative to a convex set of equivalent measures is introduced and the necessary and sufficient conditions of the local regularity of it in the discrete case are founded. The description of all local regular super-martingales relative to a convex set of equivalent measures is presented. The notion of the complete set of equivalent measures is introduced. We prove that every bounded in some sense super-martingale relative to the complete set of equivalent measures is local regular. A new definition of the fair price of contingent claim in an incomplete market is given and the formula for the fair price of Standard Option of European type is found. The proved Theorems are the generalization of the famous Doob decomposition for super-martingale onto the case of super-martingales relative to a convex set of equivalent measures.

Suggested Citation

  • Nicholas S. Gonchar, 2018. "Martingales and Super-martingales Relative to a Convex Set of Equivalent Measures," Papers 1806.05557, arXiv.org.
    • Handle: RePEc:arx:papers:1806.05557
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    References listed on IDEAS

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    1. J. Michael Harrison & Stanley R. Pliska, 1981. "Martingales and Stochastic Integrals in the Theory of Continous Trading," Discussion Papers 454, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. 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.
    3. Kreps, David M., 1981. "Arbitrage and equilibrium in economies with infinitely many commodities," Journal of Mathematical Economics, Elsevier, vol. 8(1), pages 15-35, March.
    4. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and duality in nondominated discrete-time models," Papers 1305.6008, arXiv.org, revised Mar 2015.
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    Cited by:

    1. N. S. Gonchar, 2020. "Derivatives Pricing in Non-Arbitrage Market," Papers 2010.13630, arXiv.org.
    2. N. S. Gonchar, 2018. "Description of Incomplete Financial Markets for the Discrete Time Evolution of Risk Assets," Papers 1810.09366, arXiv.org.

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