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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 H1‐martingale to be representable with respect to a collection, of local martingales. M H1(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).

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, October.
    • Handle: RePEc:bla:mathfi:v:2:y:1992:i:4:p:239-250
    DOI: 10.1111/j.1467-9965.1992.tb00031.x
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    Cited by:

    1. Abdelkarem Berkaoui, 2018. "On the degree of incompleteness of an incomplete financial market," Papers 1811.07509, arXiv.org.
    2. Berkaoui, Abdelkarem, 2023. "On the optional and orthogonal decompositions of supermartingales and applications," Statistics & Probability Letters, Elsevier, vol. 199(C).
    3. 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.
    4. Domenico Cuoco & Hua He, 2001. "Dynamic Aggregation and Computation of Equilibria in Finite-Dimensional Economies with Incomplete Financial Markets," Annals of Economics and Finance, Society for AEF, vol. 2(2), pages 265-296, November.
    5. Elyès Jouini & Hédi Kallal, 1999. "Viability and Equilibrium in Securities Markets with Frictions," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 275-292, July.
    6. repec:zbw:bofrdp:2008_005 is not listed on IDEAS
    7. Huhtala, Heli, 2008. "Along but beyond mean-variance: Utility maximization in a semimartingale model," Bank of Finland Research Discussion Papers 5/2008, Bank of Finland.
    8. Huhtala, Heli, 2008. "Along but beyond mean-variance : Utility maximization in a semimartingale model," Research Discussion Papers 5/2008, Bank of Finland.
    9. 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.
    10. Mehdi Vazifedan & Qiji Jim Zhu, 2020. "No-Arbitrage Principle in Conic Finance," Risks, MDPI, vol. 8(2), pages 1-34, June.
    11. Walter Schachermayer, 1993. "A Counterexample to Several Problems In the Theory of Asset Pricing," Mathematical Finance, Wiley Blackwell, vol. 3(2), pages 217-229, April.
    12. Martin HERDEGEN & Martin SCHWEIZER, 2016. "Economically Consistent Valuations and Put-Call Parity," Swiss Finance Institute Research Paper Series 16-02, Swiss Finance Institute.
    13. Johannes Ruf, 2010. "Hedging under arbitrage," Papers 1003.4797, arXiv.org, revised May 2011.
    14. 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.
    15. Barbachan, José Fajardo, 2002. "Equilibrium in stochastic economies with incomplete financial markets," Brazilian Review of Econometrics, Sociedade Brasileira de Econometria - SBE, vol. 22(1), May.
    16. Martin Herdegen & Martin Schweizer, 2018. "Semi‐efficient valuations and put‐call parity," Mathematical Finance, Wiley Blackwell, vol. 28(4), pages 1061-1106, October.
    17. Sergio Pulido, 2010. "The fundamental theorem of asset pricing, the hedging problem and maximal claims in financial markets with short sales prohibitions," Papers 1012.3102, arXiv.org, revised Jan 2014.
    18. Saul Jacka & Abdel Berkaoui, 2006. "On decomposing risk in a financial-intermediate market and reserving," Papers math/0603041, arXiv.org.
    19. Dragana Valjarević, 2024. "Concepts of Statistical Causality and Strong and Weak Properties of Predictable Representation," Mathematics, MDPI, vol. 12(5), pages 1-11, February.
    20. Kuhn, Christoph, 2004. "Game contingent claims in complete and incomplete markets," Journal of Mathematical Economics, Elsevier, vol. 40(8), pages 889-902, December.
    21. repec:dau:papers:123456789/5603 is not listed on IDEAS

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