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Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices

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  • Erhan Bayraktar
  • Xiang Yu

Abstract

This paper studies the utility maximization on the terminal wealth with random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system (CPS) such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility maximization problem are established using the duality approach. As an important application of the duality theorem, we provide some sufficient conditions for the existence of a shadow price process with random endowments in a generalized form as well as in the usual sense using acceptable portfolios.

Suggested Citation

  • Erhan Bayraktar & Xiang Yu, 2015. "Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices," Papers 1504.00310, arXiv.org, revised Aug 2018.
    • Handle: RePEc:arx:papers:1504.00310
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    References listed on IDEAS

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    1. Luciano Campi & Walter Schachermayer, 2006. "A super-replication theorem in Kabanov’s model of transaction costs," Finance and Stochastics, Springer, vol. 10(4), pages 579-596, December.
    2. repec:dau:papers:123456789/5455 is not listed on IDEAS
    3. Julien Hugonnier & Dmitry Kramkov, 2004. "Optimal investment with random endowments in incomplete markets," Papers math/0405293, arXiv.org.
    4. Walter Schachermayer, 2013. "Admissible Trading Strategies under Transaction Costs," Papers 1308.1492, arXiv.org, revised May 2014.
    5. Walter Schachermayer, 2004. "The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 19-48, January.
    6. Walter Schachermayer, 2014. "The super-replication theorem under proportional transaction costs revisited," Papers 1405.1266, arXiv.org.
    7. (**), 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.
    8. repec:dau:papers:123456789/5980 is not listed on IDEAS
    9. Yuri M. Kabanov & Günter Last, 2002. "Hedging under Transaction Costs in Currency Markets: a Continuous‐Time Model," Mathematical Finance, Wiley Blackwell, vol. 12(1), pages 63-70, January.
    10. Saul Jacka & Abdelkarem Berkaoui, 2006. "On the density of properly maximal claims in financial markets with transaction costs," Papers math/0602592, arXiv.org, revised May 2007.
    11. Xiang Yu, 2014. "Optimal Consumption under Habit Formation In Markets with Transaction Costs and Random Endowments," Papers 1408.1382, arXiv.org, revised Jul 2016.
    12. Julien Hugonnier & Dmitry Kramkov & Walter Schachermayer, 2005. "On Utility‐Based Pricing Of Contingent Claims In Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 203-212, April.
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    Cited by:

    1. Lingqi Gu & Yiqing Lin & Junjian Yang, 2016. "On the existence of shadow prices for optimal investment with random endowment," Papers 1602.01109, arXiv.org, revised Feb 2017.
    2. Shuoqing Deng & Xiaolu Tan & Xiang Yu, 2020. "Utility Maximization with Proportional Transaction Costs Under Model Uncertainty," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1210-1236, November.

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