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A Martingale Representation Result and an Application to Incomplete Financial Markets

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  • S. D. Jacka
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    Abstract

    We establish necessary and sufficient conditions for an H-martingale to be representable with respect to a collection, of local martingales. "M" H("P") is representable if and only if "M" is a local martingale under all p.m.'s "Q" which are "uniformly equivalent" to "P" and which make all the elements of local martingales (Theorem 1.1). We then give necessary and sufficient conditions which are easier to verify, and only involve expectations (Theorem 1.2). We go on to apply these results to the problem of pricing claims in an incomplete financial market-establishing two conjectures of Harrison and Pliska(1981). Copyright 1992 Blackwell Publishers.

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    Bibliographic Info

    Article provided by Wiley Blackwell in its journal Mathematical Finance.

    Volume (Year): 2 (1992)
    Issue (Month): 4 ()
    Pages: 239-250

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    Handle: RePEc:bla:mathfi:v:2:y:1992:i:4:p:239-250

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    Cited by:
    1. Walter Schachermayer, 1993. "A Counterexample to Several Problems In the Theory of Asset Pricing," Mathematical Finance, Wiley Blackwell, vol. 3(2), pages 217-229.
    2. Huhtala, Heli, 2008. "Along but beyond mean-variance: Utility maximization in a semimartingale model," Research Discussion Papers 5/2008, Bank of Finland.
    3. Kuhn, Christoph, 2002. "Pricing contingent claims in incomplete markets when the holder can choose among different payoffs," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 215-233, October.
    4. Johannes Ruf, 2010. "Hedging under arbitrage," Papers 1003.4797, arXiv.org, revised May 2011.
    5. Kuhn, Christoph, 2004. "Game contingent claims in complete and incomplete markets," Journal of Mathematical Economics, Elsevier, vol. 40(8), pages 889-902, December.
    6. Jouini, Elyès & Kallal, Hedi, 1999. "Viability and equilibrium in securities markets with frictions," Economics Papers from University Paris Dauphine 123456789/5603, Paris Dauphine University.
    7. Saul Jacka & Abdel Berkaoui, 2006. "On decomposing risk in a financial-intermediate market and reserving," Papers math/0603041, arXiv.org.

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