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A Martingale Representation Result and an Application to Incomplete Financial Markets

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  • S. D. Jacka

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.

Suggested Citation

  • S. D. Jacka, 1992. "A Martingale Representation Result and an Application to Incomplete Financial Markets," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 239-250.
  • 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, pages 217-229.
    2. Martin HERDEGEN & Martin SCHWEIZER, 2016. "Economically Consistent Valuations and Put-Call Parity," Swiss Finance Institute Research Paper Series 16-02, Swiss Finance Institute.
    3. Johannes Ruf, 2010. "Hedging under arbitrage," Papers 1003.4797, arXiv.org, revised May 2011.
    4. Tahir CHOULLI & Martin SCHWEIZER, 2015. "Locally Phi-Integrable Sigma-Martingale Densities for General Semimartingales," Swiss Finance Institute Research Paper Series 15-15, Swiss Finance Institute.
    5. Kuhn, Christoph, 2002. "Pricing contingent claims in incomplete markets when the holder can choose among different payoffs," Insurance: Mathematics and Economics, Elsevier, pages 215-233.
    6. repec:sbe:breart:v:22:y:2002:i:1:a:2745 is not listed on IDEAS
    7. Saul Jacka & Abdel Berkaoui, 2006. "On decomposing risk in a financial-intermediate market and reserving," Papers math/0603041, arXiv.org.
    8. Kuhn, Christoph, 2004. "Game contingent claims in complete and incomplete markets," Journal of Mathematical Economics, Elsevier, vol. 40(8), pages 889-902, December.
    9. repec:dau:papers:123456789/5603 is not listed on IDEAS
    10. Huhtala, Heli, 2008. "Along but beyond mean-variance : Utility maximization in a semimartingale model," Research Discussion Papers 5/2008, Bank of Finland.
    11. Sven Rady, 1997. "Option pricing in the presence of natural boundaries and a quadratic diffusion term (*)," Finance and Stochastics, Springer, vol. 1(4), pages 331-344.

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