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Backward Stochastic PDEs Related to the Utility Maximization Problem

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  • Michael Mania

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  • Revaz Tevzadze

    ()

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    Abstract

    We study utility maximization problem for general utility functions using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an Rd-valued continuous semimartingale. Under some regularity assumptions we derive backward stochastic partial differential equation (BSPDE) related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As examples the cases of power, exponential and logarithmic utilities are considered

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    File URL: http://www.icer.it/docs/wp2008/ICERwp07-08.pdf
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    Bibliographic Info

    Paper provided by ICER - International Centre for Economic Research in its series ICER Working Papers - Applied Mathematics Series with number 07-2008.

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    Length: 25 pages
    Date of creation: Jun 2008
    Date of revision:
    Handle: RePEc:icr:wpmath:07-2008

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    Keywords: Backward stochastic partial di erential equation; utility maximization problem; semimartingale; incomplete markets;

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    1. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71.
    2. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52.
    3. (**), Hui Wang & Jaksa Cvitanic & (*), Walter Schachermayer, 2001. "Utility maximization in incomplete markets with random endowment," Finance and Stochastics, Springer, vol. 5(2), pages 259-272.
    4. Dmitry Kramkov & Mihai S\^{{\i}}rbu, 2006. "On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets," Papers math/0610224, arXiv.org.
    5. David Heath & Eckhard Platen & Martin Schweizer, 2001. "A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 11(4), pages 385-413.
    6. Yuri M. Kabanov & Christophe Stricker, 2002. "On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 125-134.
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    Cited by:
    1. Imkeller, Peter & Reveillac, Anthony & Zhang, Jianing, 2011. "Solvability and numerical simulation of BSDEs related to BSPDEs with applications to utility maximization," Economics Papers from University Paris Dauphine 123456789/7101, Paris Dauphine University.
    2. Claudia Ceci & Anna Gerardi, 2011. "Utility indifference valuation for jump risky assets," Decisions in Economics and Finance, Springer, vol. 34(2), pages 85-120, November.

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