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Stability of utility-maximization in incomplete markets

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  • Larsen, Kasper
  • Zitkovic, Gordan
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    Abstract

    The effectiveness of utility-maximization techniques for portfolio management relies on our ability to estimate correctly the parameters of the dynamics of the underlying financial assets. In the setting of complete or incomplete financial markets, we investigate whether small perturbations of the market coefficient processes lead to small changes in the agent's optimal behavior, as derived from the solution of the related utility-maximization problems. Specifically, we identify the topologies on the parameter process space and the solution space under which utility-maximization is a continuous operation, and we provide a counterexample showing that our results are best possible, in a certain sense. A novel result about the structure of the solution of the utility-maximization problem, where prices are modeled by continuous semimartingales, is established as an offshoot of the proof of our central theorem.

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    File URL: http://www.sciencedirect.com/science/article/B6V1B-4NWCH41-4/2/576ed4ff221c1a9d52692b021124d71c
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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 117 (2007)
    Issue (Month): 11 (November)
    Pages: 1642-1662

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    Handle: RePEc:eee:spapps:v:117:y:2007:i:11:p:1642-1662

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    Related research

    Keywords: Appropriate topologies Continuous semimartingales Convex duality Market price of risk process Mathematical finance Utility-maximization V-relative compactness Well-posedness;

    References

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    1. He, Hua & Pearson, Neil D., 1991. "Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case," Journal of Economic Theory, Elsevier, vol. 54(2), pages 259-304, August.
    2. Cox, John C. & Huang, Chi-fu, 1989. "Optimal consumption and portfolio policies when asset prices follow a diffusion process," Journal of Economic Theory, Elsevier, vol. 49(1), pages 33-83, October.
    3. Clotilde Napp & Elyès Jouini, 2004. "Convergence of utility functions and convergence of optimal strategies," Post-Print halshs-00151579, HAL.
    4. L.C.G. Rogers, 2001. "The relaxed investor and parameter uncertainty," Finance and Stochastics, Springer, vol. 5(2), pages 131-154.
    5. Friedrich Hubalek & Walter Schachermayer, 1998. "When Does Convergence of Asset Price Processes Imply Convergence of Option Prices?," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 385-403.
    6. Freddy Delbaen & Walter Schachermayer, 1998. "A Simple Counterexample to Several Problems in the Theory of Asset Pricing," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 1-11.
    7. Nicole El Karoui & Monique Jeanblanc-Picquè & Steven E. Shreve, 1998. "Robustness of the Black and Scholes Formula," Mathematical Finance, Wiley Blackwell, vol. 8(2), pages 93-126.
    8. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    9. Napp, Clotilde & Jouini, Elyès, 2004. "Convergence of utility functions and convergence of optimal strategies," Economics Papers from University Paris Dauphine 123456789/355, Paris Dauphine University.
    10. R. C. Merton, 1970. "Optimum Consumption and Portfolio Rules in a Continuous-time Model," Working papers 58, Massachusetts Institute of Technology (MIT), Department of Economics.
    11. Trojani, Fabio & Vanini, Paolo, 2002. "A note on robustness in Merton's model of intertemporal consumption and portfolio choice," Journal of Economic Dynamics and Control, Elsevier, vol. 26(3), pages 423-435, March.
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    Citations

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    Cited by:
    1. Horst, Ulrich & Hu, Ying & Imkeller, Peter & Réveillac, Anthony & Zhang, Jianing, 2014. "Forward–backward systems for expected utility maximization," Stochastic Processes and their Applications, Elsevier, vol. 124(5), pages 1813-1848.
    2. Bayraktar, Erhan & Kravitz, Ross, 2013. "Stability of exponential utility maximization with respect to market perturbations," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1671-1690.
    3. Michail Anthropelos & Gordan Žitković, 2010. "Partial equilibria with convex capital requirements: existence, uniqueness and stability," Annals of Finance, Springer, vol. 6(1), pages 107-135, January.
    4. Gordan Žitković, 2012. "An example of a stochastic equilibrium with incomplete markets," Finance and Stochastics, Springer, vol. 16(2), pages 177-206, April.
    5. Constantinos Kardaras, 2008. "The continuous behavior of the numeraire portfolio under small changes in information structure, probabilistic views and investment constraints," Papers 0804.2912, arXiv.org, revised Nov 2009.
    6. Ulrich Horst & Ying Hu & Peter Imkeller & Anthony Reveillac, 2011. "Forward-backward systems for expected utility maximization," SFB 649 Discussion Papers SFB649DP2011-061, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    7. Pietro Siorpaes, 2013. "Optimal investment and price dependence in a semi-static market," Papers 1303.0237, arXiv.org, revised Oct 2013.
    8. Michail Anthropelos & Gordan Zitkovic, 2009. "Partial Equilibria with Convex Capital Requirements: Existence, Uniqueness and Stability," Papers 0901.3318, arXiv.org.
    9. Gordan Zitkovic, 2009. "An example of a stochastic equilibrium with incomplete markets," Papers 0906.0208, arXiv.org, revised Jun 2010.
    10. Frei, Christoph & Mocha, Markus & Westray, Nicholas, 2012. "BSDEs in utility maximization with BMO market price of risk," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2486-2519.
    11. Kardaras, Constantinos, 2010. "The continuous behavior of the numéraire portfolio under small changes in information structure, probabilistic views and investment constraints," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 331-347, March.
    12. Christoph Frei & Markus Mocha & Nicholas Westray, 2011. "BSDEs in Utility Maximization with BMO Market Price of Risk," Papers 1107.0183, arXiv.org, revised Feb 2012.
    13. Kasper Larsen & Hang Yu, 2012. "Horizon dependence of utility optimizers in incomplete models," Finance and Stochastics, Springer, vol. 16(4), pages 779-801, October.
    14. Paolo Guasoni & Constantinos Kardaras & Scott Robertson & Hao Xing, 2014. "Abstract, classic, and explicit turnpikes," Finance and Stochastics, Springer, vol. 18(1), pages 75-114, January.
    15. Markus Mocha & Nicholas Westray, 2011. "The Stability of the Constrained Utility Maximization Problem - A BSDE Approach," Papers 1107.0190, arXiv.org.

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