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BSDEs in Utility Maximization with BMO Market Price of Risk

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  • Christoph Frei
  • Markus Mocha
  • Nicholas Westray
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    Abstract

    This article studies quadratic semimartingale BSDEs arising in power utility maximization when the market price of risk is of BMO type. In a Brownian setting we provide a necessary and sufficient condition for the existence of a solution but show that uniqueness fails to hold in the sense that there exists a continuum of distinct square-integrable solutions. This feature occurs since, contrary to the classical Ito representation theorem, a representation of random variables in terms of stochastic exponentials is not unique. We study in detail when the BSDE has a bounded solution and derive a new dynamic exponential moments condition which is shown to be the minimal sufficient condition in a general filtration. The main results are complemented by several interesting examples which illustrate their sharpness as well as important properties of the utility maximization BSDE.

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    File URL: http://arxiv.org/pdf/1107.0183
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1107.0183.

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    Date of creation: Jul 2011
    Date of revision: Feb 2012
    Publication status: Published in Stochastic Process. Appl., 122 (6): 2486 - 2519, 2012
    Handle: RePEc:arx:papers:1107.0183

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    1. Marie-Amélie Morlais, 2009. "Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem," Finance and Stochastics, Springer, Springer, vol. 13(1), pages 121-150, January.
    2. Marcel Nutz, 2009. "The Opportunity Process for Optimal Consumption and Investment with Power Utility," Papers 0912.1879, arXiv.org, revised Jun 2010.
    3. Kasper Larsen & Gordan Zitkovic, 2007. "Stability of utility-maximization in incomplete markets," Papers 0706.0474, arXiv.org.
    4. Marcel Nutz, 2010. "Risk Aversion Asymptotics for Power Utility Maximization," Papers 1003.3582, arXiv.org.
    5. Michael Mania & Martin Schweizer, 2005. "Dynamic exponential utility indifference valuation," Papers math/0508489, arXiv.org.
    6. Richard Rouge & Nicole El Karoui, 2000. "Pricing Via Utility Maximization and Entropy," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 10(2), pages 259-276.
    7. Larsen, Kasper & Zitkovic, Gordan, 2007. "Stability of utility-maximization in incomplete markets," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 117(11), pages 1642-1662, November.
    8. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-57, August.
    9. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
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