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Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs

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  • Jin Hyuk Choi
  • Mihai Sirbu
  • Gordan Zitkovic
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    Abstract

    We revisit the optimal investment and consumption model of Davis and Norman (1990) and Shreve and Soner (1994), following a shadow-price approach similar to that of Kallsen and Muhle-Karbe (2010). Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton-Jacobi-Bellman equation for this singular stochastic control problem to a non-standard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. Furthermore, we provide an explicit characterization of model parameters for which the value function is finite.

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    File URL: http://arxiv.org/pdf/1204.0305
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1204.0305.

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    Date of creation: Apr 2012
    Date of revision: Jun 2012
    Handle: RePEc:arx:papers:1204.0305

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    Web page: http://arxiv.org/

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    1. Magill, Michael J. P. & Constantinides, George M., 1976. "Portfolio selection with transactions costs," Journal of Economic Theory, Elsevier, Elsevier, vol. 13(2), pages 245-263, October.
    2. Victor Ginsburgh & Michiel Keyzer, 2002. "The structure of applied general equilibrium models," ULB Institutional Repository 2013/3313, ULB -- Universite Libre de Bruxelles.
    3. Jouini Elyes & Kallal Hedi, 1995. "Martingales and Arbitrage in Securities Markets with Transaction Costs," Journal of Economic Theory, Elsevier, Elsevier, vol. 66(1), pages 178-197, June.
    4. J. Kallsen & J. Muhle-Karbe, 2010. "On using shadow prices in portfolio optimization with transaction costs," Papers 1010.4989, arXiv.org.
    5. repec:fth:inseep:9513 is not listed on IDEAS
    6. Kallal, Hedi & Jouini, Elyès, 1995. "Martingales and arbitrage in securities markets with transaction costs," Economics Papers from University Paris Dauphine, Paris Dauphine University 123456789/5630, Paris Dauphine University.
    7. repec:wop:humbsf:1998-18 is not listed on IDEAS
    8. Lamberton, Damien & Pham, Huyên & Schweizer, Martin, 1998. "Local risk-minimization under transaction costs," SFB 373 Discussion Papers 1998,18, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
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    Cited by:
    1. Dmitry Rokhlin, 2013. "On the game interpretation of a shadow price process in utility maximization problems under transaction costs," Finance and Stochastics, Springer, Springer, vol. 17(4), pages 819-838, October.
    2. Paolo Guasoni & Johannes Muhle-Karbe, 2012. "Portfolio Choice with Transaction Costs: a User's Guide," Papers 1207.7330, arXiv.org.

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