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Super-replication and utility maximization in large financial markets

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  • De Donno, M.
  • Guasoni, P.
  • Pratelli, M.

Abstract

We study the problems of super-replication and utility maximization from terminal wealth in a semimartingale model with countably many assets. After introducing a suitable definition of admissible strategy, we characterize superreplicable contingent claims in terms of martingale measures. Utility maximization problems are then studied with the convex duality method, and we extend finite-dimensional results to this setting. The existence of an optimizer is proved in a suitable class of generalized strategies: this class has also the property that maximal expected utility is the limit of maximal expected utilities in finite-dimensional submarkets. Finally, we illustrate our results with some examples in infinite dimensional factor models.

Suggested Citation

  • De Donno, M. & Guasoni, P. & Pratelli, M., 2005. "Super-replication and utility maximization in large financial markets," Stochastic Processes and their Applications, Elsevier, vol. 115(12), pages 2006-2022, December.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:12:p:2006-2022
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    References listed on IDEAS

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    1. Irene Klein, 2000. "A Fundamental Theorem of Asset Pricing for Large Financial Markets," Mathematical Finance, Wiley Blackwell, vol. 10(4), pages 443-458, October.
    2. Stephen A. Ross, 2013. "The Arbitrage Theory of Capital Asset Pricing," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 1, pages 11-30, World Scientific Publishing Co. Pte. Ltd..
    3. Marzia De Donno, 2004. "A note on completeness in large financial markets," Mathematical Finance, Wiley Blackwell, vol. 14(2), pages 295-315, April.
    4. Y.M. Kabanov & D.O. Kramkov, 1998. "Asymptotic arbitrage in large financial markets," Finance and Stochastics, Springer, vol. 2(2), pages 143-172.
    5. Gur Huberman, 2005. "A Simple Approach to Arbitrage Pricing Theory," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 9, pages 289-308, World Scientific Publishing Co. Pte. Ltd..
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    Citations

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    Cited by:

    1. M. De Donno & M. Pratelli, 2006. "A theory of stochastic integration for bond markets," Papers math/0602532, arXiv.org.
    2. Miklós Rásonyi, 2016. "On Optimal Strategies For Utility Maximizers In The Arbitrage Pricing Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(07), pages 1-12, November.
    3. Yushi Hamaguchi, 2018. "BSDEs driven by cylindrical martingales with application to approximate hedging in bond markets," Papers 1806.04025, arXiv.org.
    4. Scott Robertson & Konstantinos Spiliopoulos, 2014. "Indifference pricing for Contingent Claims: Large Deviations Effects," Papers 1410.0384, arXiv.org, revised Feb 2016.
    5. Laurence Carassus & Miklos Rasonyi, 2019. "From small markets to big markets," Papers 1907.05593, arXiv.org, revised Oct 2020.
    6. Christa Cuchiero & Irene Klein & Josef Teichmann, 2017. "A fundamental theorem of asset pricing for continuous time large financial markets in a two filtration setting," Papers 1705.02087, arXiv.org.
    7. Constantinos Kardaras, 2011. "On the closure in the Emery topology of semimartingale wealth-process sets," Papers 1108.0945, arXiv.org, revised Jul 2013.
    8. Mahan Tahvildari, 2021. "Forward indifference valuation and hedging of basis risk under partial information," Papers 2101.00251, arXiv.org.
    9. Constantinos Kardaras, 2019. "Stochastic integration with respect to arbitrary collections of continuous semimartingales and applications to Mathematical Finance," Papers 1908.03946, arXiv.org, revised Aug 2019.
    10. Laurence Carassus & Miklós Rásonyi, 2020. "Risk-Neutral Pricing for Arbitrage Pricing Theory," Journal of Optimization Theory and Applications, Springer, vol. 186(1), pages 248-263, July.
    11. Winslow Strong, 2011. "Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension," Papers 1112.5340, arXiv.org.
    12. Winslow Strong, 2014. "Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension," Finance and Stochastics, Springer, vol. 18(3), pages 487-514, July.
    13. Christa Cuchiero & Irene Klein & Josef Teichmann, 2014. "A new perspective on the fundamental theorem of asset pricing for large financial markets," Papers 1412.7562, arXiv.org, revised Oct 2023.
    14. Miklos Rasonyi, 2015. "Maximizing expected utility in the Arbitrage Pricing Model," Papers 1508.07761, arXiv.org, revised Mar 2017.
    15. Oleksii Mostovyi, 2014. "Utility maximization in the large markets," Papers 1403.6175, arXiv.org, revised Oct 2014.
    16. Miklos Rasonyi, 2017. "On utility maximization without passing by the dual problem," Papers 1702.00982, arXiv.org, revised Mar 2018.

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