Finance and Stochastics

, Volume 20, Issue 1, pp 99–121 | Cite as

Facelifting in utility maximization

  • Kasper Larsen
  • Halil Mete Soner
  • Gordan Žitković


We establish the existence and characterization of a primal and a dual facelift—discontinuity of the value function at the terminal time—for utility maximization in incomplete semimartingale-driven financial markets. Unlike in the lower and upper hedging problems, and somewhat unexpectedly, a facelift turns out to exist in utility maximization despite strict convexity in the objective function. In addition to discussing our results in their natural, Markovian environment, we also use them to show that the dual optimizer cannot be found in the set of countably additive (martingale) measures in a wide variety of situations.


Boundary layer Convex analysis Convex duality Facelift Financial mathematics Incomplete markets Markov processes Utility maximization Unspanned endowment 

Mathematics Subject Classification (2010)

91G10 91G80 60K35 

JEL Classification

C61 G11 



The authors would like to thank Mihai Sîrbu, Kim Weston, and the two anonymous referees for their many constructive comments.


  1. 1.
    Ansel, J.-P., Stricker, C.: Couverture des actifs contingents et prix maximum. Ann. Inst. Henri Poincaré Probab. Stat. 30, 303–315 (1994) MATHMathSciNetGoogle Scholar
  2. 2.
    Broadie, M., Cvitanić, J., Soner, H.M.: Optimal replication of contingent claims under portfolio constraints. Rev. Financ. Stud. 11, 59–79 (1998) CrossRefGoogle Scholar
  3. 3.
    Bouchard, B., Touzi, N.: Explicit solution to the multivariate super-replication problem under transaction costs. Ann. Appl. Probab. 10, 685–708 (2000) MATHMathSciNetGoogle Scholar
  4. 4.
    Chassagneux, J.-F., Elie, R., Kharroubi, I.: When terminal facelift enforces delta constraints. Finance Stoch. 19, 329–362 (2015) MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Cvitanić, J., Schachermayer, W., Wang, H.: Utility maximization in incomplete markets with random endowment. Finance Stoch. 5, 237–259 (2001) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250 (1998) MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Guasoni, P., Rásonyi, M., Schachermayer, W.: Consistent price systems and face-lifting pricing under transaction costs. Ann. Appl. Probab. 18, 491–520 (2008) MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Hugonnier, J., Kramkov, D., Schachermayer, W.: On utility based pricing of contingent claims in incomplete markets. Math. Finance 15, 203–212 (2005) MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Kabanov, Y., Stricker, C.: On equivalent martingale measures with bounded densities. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds.) Séminaire de Probabilités XXXV. Lecture Notes in Math., vol. 1755, pp. 139–148 (2001) CrossRefGoogle Scholar
  10. 10.
    Kramkov, D.: Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Relat. Fields 105, 459–479 (1996) MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999) MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, Berlin (1998) MATHCrossRefGoogle Scholar
  13. 13.
    Karatzas, I., Žitković, G.: Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31, 1821–1858 (2003) MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Larsen, K., Žitković, G.: Utility maximization under convex portfolio constraints. Ann. Appl. Probab. 23, 665–692 (2013) MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Pham, H.: Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer, Berlin (2009) MATHCrossRefGoogle Scholar
  16. 16.
    Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Applications of Mathematics (New York), vol. 21. Springer, Berlin (2004) MATHGoogle Scholar
  17. 17.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer, Berlin (1998) MATHGoogle Scholar
  18. 18.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293. Springer, Berlin (1999) MATHCrossRefGoogle Scholar
  19. 19.
    Schmock, U., Shreve, S.E., Wystup, U.: Valuation of exotic options under shortselling constraints. Finance Stoch. 6, 143–172 (2002) MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Soner, H.M., Touzi, N.: Superreplication under gamma constraints. SIAM J. Control Optim. 39, 73–96 (2000) MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Soner, H.M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4, 201–236 (2002) MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Touzi, N.: Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Fields Institute Monographs, vol. 29. Springer, New York (2013) MATHCrossRefGoogle Scholar
  23. 23.
    Žitković, G.: Dynamic programming for controlled Markov families: abstractly and over martingale measures. SIAM J. Control Optim. 52, 1597–1621 (2014) MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Kasper Larsen
    • 1
  • Halil Mete Soner
    • 2
  • Gordan Žitković
    • 3
  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of MathematicsETHZürichSwitzerland
  3. 3.Department of MathematicsUniversity of Texas at AustinAustinUSA

Personalised recommendations