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Multivariate utility maximization with proportional transaction costs and random endowment


  • Giuseppe Benedetti

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris-Dauphine - CNRS - Centre National de la Recherche Scientifique, CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - ENSAE ParisTech - École Nationale de la Statistique et de l'Administration Économique)

  • Luciano Campi

    () (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris-Dauphine - CNRS - Centre National de la Recherche Scientifique, FiME Lab - Laboratoire de Finance des Marchés d'Energie - Université Paris-Dauphine - CREST - EDF R&D - EDF R&D - EDF - EDF)


In this paper we deal with a utility maximization problem at finite horizon on a continuous-time market with conical (and time varying) constraints (particularly suited to model a currency market with proportional transaction costs). In particular, we extend the results in Campi and Owen (2011) to the situation where the agent is initially endowed with a random and possibly unbounded quantity of assets. We start by studying some basic properties of the value function (which is now defined on a space of random variables), then we dualize the problem following some convex analysis techniques which have proven very useful in this field of research. We finally prove the existence of a solution to the dual and (under an additional boundedness assumption on the endowment) to the primal problem. The last section of the paper is devoted to an application of our results to utility indifference pricing.

Suggested Citation

  • Giuseppe Benedetti & Luciano Campi, 2011. "Multivariate utility maximization with proportional transaction costs and random endowment," Working Papers hal-00586377, HAL.
  • Handle: RePEc:hal:wpaper:hal-00586377
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    References listed on IDEAS

    1. Paolo Guasoni & Emmanuel Lépinette & Miklós Rásonyi, 2012. "The fundamental theorem of asset pricing under transaction costs," Finance and Stochastics, Springer, vol. 16(4), pages 741-777, October.
    2. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
    3. Elyés Jouini & Moncef Meddeb & Nizar Touzi, 2004. "Vector-valued coherent risk measures," Finance and Stochastics, Springer, vol. 8(4), pages 531-552, November.
    4. Claus Munk, 1999. "The Valuation of Contingent Claims under Portfolio Constraints: Reservation Buying and Selling Prices," Review of Finance, European Finance Association, vol. 3(3), pages 347-388.
    5. (**), 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.
    6. Jouini Elyes & Kallal Hedi, 1995. "Martingales and Arbitrage in Securities Markets with Transaction Costs," Journal of Economic Theory, Elsevier, vol. 66(1), pages 178-197, June.
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    Cited by:

    1. Christoph Czichowsky & Walter Schachermayer, 2015. "Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion," Papers 1505.02416,, revised Aug 2016.
    2. Christoph Czichowsky & Walter Schachermayer, 2014. "Duality Theory for Portfolio Optimisation under Transaction Costs," Papers 1408.5989,
    3. Teemu Pennanen & Ari-Pekka Perkkio, 2016. "Convex duality in optimal investment and contingent claim valuation in illiquid markets," Papers 1603.02867,
    4. Christoph Czichowsky & R'emi Peyre & Walter Schachermayer & Junjian Yang, 2016. "Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs," Papers 1608.01415,
    5. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the Existence of Shadow Prices," Papers 1111.6633,, revised Jan 2013.
    6. Czichowsky, Christoph & Schachermayer, Walter, 2016. "Duality theory for portfolio optimisation under transaction costs," LSE Research Online Documents on Economics 63362, London School of Economics and Political Science, LSE Library.
    7. Jörn Sass & Martin Smaga, 2014. "FTAP in finite discrete time with transaction costs by utility maximization," Finance and Stochastics, Springer, vol. 18(4), pages 805-823, October.
    8. Miklos Rasonyi, 2017. "On utility maximization without passing by the dual problem," Papers 1702.00982,, revised Mar 2018.
    9. Christoph Czichowsky & Johannes Muhle-Karbe & Walter Schachermayer, 2012. "Transaction Costs, Shadow Prices, and Duality in Discrete Time," Papers 1205.4643,, revised Jan 2014.
    10. Huy N. Chau & Mikl'os R'asonyi, 2016. "Skorohod's representation theorem and optimal strategies for markets with frictions," Papers 1606.07311,, revised Apr 2017.
    11. Lingqi Gu & Yiqing Lin & Junjian Yang, 2017. "Utility maximization problem under transaction costs: optimal dual processes and stability," Papers 1710.04363,
    12. Yiqing Lin & Junjian Yang, 2016. "Utility maximization problem with random endowment and transaction costs: when wealth may become negative," Papers 1604.08224,, revised Sep 2016.
    13. repec:wsi:ijtafx:v:20:y:2017:i:05:n:s0219024917500261 is not listed on IDEAS
    14. Lingqi Gu & Yiqing Lin & Junjian Yang, 2016. "A note on utility maximization with transaction costs and random endoment: num\'eraire-based model and convex duality," Papers 1602.01070,, revised Feb 2016.
    15. repec:spr:finsto:v:22:y:2018:i:1:d:10.1007_s00780-017-0351-5 is not listed on IDEAS
    16. Giuseppe Benedetti & Luciano Campi & Jan Kallsen & Johannes Muhle-Karbe, 2011. "On the existence of shadow prices," Working Papers hal-00645980, HAL.

    More about this item


    Transaction costs; Foreign exchange market; Multivariate utility function; Optimal portfolio; Duality theory; Random endowment; Utility-based pricing;

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