Optimal investment with random endowments and transaction costs: duality theory and shadow prices

  • Erhan Bayraktar
  • Xiang Yu


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 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 similar to Czichowsky and Schachermayer (Ann Appl Probab 26(3):1888–1941, 2016) as well as in the usual sense using acceptable portfolios.


Proportional transaction costs Unbounded random endowments Acceptable portfolios Utility maximization Convex duality Shadow prices 

JEL Classification

G11 G13 



E. Bayraktar is supported in part by the National Science Foundation under Grant DMS-1613170 and the Susan M. Smith Professorship. X. Yu is supported by the Hong Kong Early Career Scheme under Grant 25302116 and the Start-Up Fund of the Hong Kong Polytechnic University under Grant 1-ZE5A.


  1. 1.
    Bayraktar, E., Yu, X.: On the market viability under proportional transaction costs. Math. Financ. 28(3), 800–838 (2018)CrossRefzbMATHGoogle Scholar
  2. 2.
    Benedetti, G., Campi, L.: Multivariate utility maximization with proportional transaction costs and random endowment. SIAM J. Control Optim. 50(3), 1283–1308 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Campi, L., Schachermayer, W.: A super-replication theorem in kabanov’s model of transaction costs. Financ. Stoch. 10, 579–596 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cvitanić, J., Schachermayer, W., Wang, H.: Utility maximization in incomplete markets with random endowment. Financ. Stoch. 5(2), 259–272 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Czichowsky, C., Schachermayer, W.: Duality theory for portfolio optimization under transaction costs. Ann. Appl. Probab. 26(3), 1888–1941 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Czichowsky, C., Schachermayer, W.: Strong supermartingales and limits of non-negative martingales. Ann. Probab. 44(1), 171–205 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Czichowsky, C., Schachermayer, W., Yang, J.: Shadow prices for continuous processes. Math. Financ. 27(3), 623–658 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300(3), 463–520 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Delbaen, F., Schachermayer, W.: The Banach space of workable contingent claims in arbitrage theory. Ann. de l’Institut Henri Poincaré Probab. et Stat. 33(1), 113–144 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dellacherie, C., Meyer, P.: Probabilities and Potential B. Theory of Martingales. Elsevier, North-Holland (1982)zbMATHGoogle Scholar
  11. 11.
    Graves, L.: The Theory of Functions of Real Variables. McGraw-Hill, New York (1946)zbMATHGoogle Scholar
  12. 12.
    Hugonnier, J., Kramkov, D.: Optimal investment with random endowments in incomplete markets. Ann. Appl. Probab. 14(2), 845–864 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hugonnier, J., Kramkov, D., Schachermayer, W.: On utility-based pricing of contingent claims in incomplete markets. Math. Financ. 15(2), 203–212 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jacka, S., Berkaoui, A.: On the density of properly maximal claims in financial markets with transaction costs. Ann. Appl. Probab. 17(2), 716–740 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kabanov, Y., Safarian, M.: Markets with transaction costs. Springer, Berlin (2009)zbMATHGoogle Scholar
  16. 16.
    Kabanov, Y.M., Last, G.: Hedging under transaction costs in currency markets: a continuous-time model. Math. Financ. 12(1), 63–70 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kabanov, Y.M., Stricker, C.: Hedging of contingent claims under transaction costs. In: Sandmann, K., Schönbucher, P.J. (eds.) Advances in Finance and Stochastics, pp. 125–136. Springer, Berlin (2002)Google Scholar
  18. 18.
    Karatzas, I., Žitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31(4), 1821–1858 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Khasanov, R.V.: Utility maximization problem in the case of unbounded endowment. Mosc. Univ. Math. Bull. 68(3), 138–147 (2013)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9(3), 904–950 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kramkov, D., Sîrbu, M.: Sensitivity analysis of utility-based prices and risk-tolerance wealth processes. Ann. Appl. Probab. 16(4), 2140–2194 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kramkov, D., Sîrbu, M.: Asymptotic analysis of utility-based hedging strategies for small number of contingent claims. Stoch. Process. Appl. 117(11), 1606–1620 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Larsen, K., Soner, H., Žitković, G.: Conditional Davis Pricing. ArXiv Preprint, arXiv:1702.02087 (2017)
  24. 24.
    Mostovyi, O.: Optimal investment with intermediate consumption and random endowment. Math. Financ. 27(1), 96–114 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Financ. 14(1), 19–48 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Schachermayer, W.: The super-replication theorem under proportional transaction costs revisited. Math. Financ. Econ. 8(4), 383–398 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schachermayer, W.: Admissible trading strategies under transaction costs. Seminaire de Probabilite XLVI. Lecture Notes in Mathematics 2123, pp. 317–331 (2015)Google Scholar
  28. 28.
    Yu, X.: Utility maximization with addictive consumption habit formation in incomplete semimartingale markets. Ann. Appl. Probab. 25(3), 1383–1419 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yu, X.: Optimal consumption under habit formation in markets with transaction costs and random endowments. Ann. Appl. Probab. 27(2), 960–1002 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Žitković, G.: Utility maximization with a stochastic clock and an unbounded random endowment. Ann. Appl. Probab. 15(1B), 748–777 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloonHong Kong

Personalised recommendations