Skip to main content
SpringerLink
Log in

Expected Utility Maximization Problem Under State Constraints and Model Uncertainty

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We study a general robust utility maximization problem from terminal wealth and consumption under state constraints. Our framework includes financial models with constrained portfolios, labor income and large investor models. We state the existence and the uniqueness of the consumption–investment strategy by studying the associated quadratic backward stochastic differential equation. We characterize the optimal control by using the duality method and deriving a dynamic maximum principle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. von Neumann, J., Morgenstern, O.: Theory of Games and economic Behavior. 2d ed., Princeton University Press, Princeton, NJ

  2. Merton, R.: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3, 373–413 (1971)

    Article  MathSciNet  Google Scholar 

  3. Karatzas, I., Lehoczky, J.P., Shreve, S., Xu, G.L.: Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optim. 29, 702–730 (1991)

    Article  MathSciNet  Google Scholar 

  4. He, H., Pearson, N.: Consumption and portfolio policies with incomplete markets and short-selling constraints: the infinite-dimensional case. J. Econ. Theory 54, 259–304 (1991)

    Article  Google Scholar 

  5. Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)

    Article  MathSciNet  Google Scholar 

  6. Cvitanic, J., Karatzas, I.: Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2, 718–767 (1992)

    Article  MathSciNet  Google Scholar 

  7. Cuoco, D.: Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. J. Econ.Theory 72, 33–73 (1997)

    Article  MathSciNet  Google Scholar 

  8. Cvitanić, J., Schachermayer, W., Wang, H.: Utility Maximization in Incomplete Markets with Random Endowment, preprint (2000)

  9. He, H., Pagès, H.: Labor income, borrowing constraints and equilibrium asset prices. Econ. Theor. 3, 663–696 (1993)

    Article  MathSciNet  Google Scholar 

  10. El Karoui, N., Jeanblanc, M.: Optimization of consumption with labor income. Finance Stochast. 4, 409–440 (1998)

    Article  Google Scholar 

  11. Mnif, M., Pham, H.: Stochastic optimization under constraints. Stoch. Process. Appl. 93, 149–180 (2001)

    Article  MathSciNet  Google Scholar 

  12. Knight, F.: Risk: Uncertainty and Profit. Houghton Mifflin, Boston (1921)

    Google Scholar 

  13. Föllmer, H., Shied, A., Weber, S.: Robust preferences and robust portfolio choice. Handb. Numer. Anal. 15, 29–87 (2009)

    MATH  Google Scholar 

  14. Maccheroni, F., Marinacci, M., Rustichini, A.: Ambiguity aversion, robustness and the variational representation of preferences. Econometrica 74, 1447–1498 (2006)

    Article  MathSciNet  Google Scholar 

  15. Quenez, M.Q.: Optimal portfolio in a multiple-priors model. In: Dalang, R., Dozzi, M., Russo, F. (eds.) Seminar on Stochastic Analysis, Random Fields and Applications IV, Progess in Probability 58, pp. 291–321. Birkhauser, Basel (2004)

    Google Scholar 

  16. Schied, A., Wu, C.T.: Duality theory for optimal investments under model uncertainty. Stat. Decis. 2, 199–217 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Anderson, E., Hansen, L.P., Sargent, T.: A quartet of semigroups for model specification, robustness, prices of risk and model detection. J. Eur. Econ. Assoc. 1, 68–123 (2003)

    Article  Google Scholar 

  18. Skiadas, C.: Robust control and recursive utility. Finance Stoch. 7, 475–489 (2003)

    Article  MathSciNet  Google Scholar 

  19. Bordigoni, G., Matoussi, A., Schweizer, M.: A stochastic control approach to a robust utility maximization problem. In: Benth, F.E. et al. (eds.), Stochastic Analysis and Applications, Proceedings of the Second Abel Symposium, Oslo, Springer, Berlin, pp. 125–151 (2007)

    Chapter  Google Scholar 

  20. Faidi, W., Matoussi, A., Mnif, M.: Maximization of recursive utilities. A dynamic maximum principle approach. SIAM J. Financ. Math. 2, 1014–1041 (2011)

    Article  MathSciNet  Google Scholar 

  21. Duffie, D., Skiadas, C.: Continuous-time security pricing: a utility gradient approach. J. Math. Econ. 23, 107–131 (1994)

    Article  MathSciNet  Google Scholar 

  22. El Karoui, N., Peng, S., Quenez, M.C.: A dynamic maximum principle for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 3, 664–693 (2001)

    MathSciNet  MATH  Google Scholar 

  23. Schroder, M., Skiadas, C.: Optimal lifetime consumption-portfolio strategies under trading constraints and generalized recursive preferences. Stoch. Process. Appl. 108, 155–202 (2003)

    Article  MathSciNet  Google Scholar 

  24. Matoussi, M., Mezghani, H., Mnif, M.: Robust utility maximization under convex portfolio constraints. Appl. Math. Optim. 71(2), 313–351 (2014)

    Article  MathSciNet  Google Scholar 

  25. Föllmer, H., Kramkov, D.: Optional decomposition under constraints. Probab. Theory Relat. Fields 109, 1–25 (1997)

    Article  MathSciNet  Google Scholar 

  26. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

  27. Pham, H.: Minimizing shortfall risk and applications to finance and insurance problems. Ann. Appl. Probab. 12(1), 143–172 (2002)

    Article  MathSciNet  Google Scholar 

  28. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Spinger, Berlin (1991)

    MATH  Google Scholar 

  29. Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)

    Article  MathSciNet  Google Scholar 

  30. Dellacherie, C., Meyer, P.A.: Probabilité et Potentiel, Chapitre I à IV. Hermann, Paris (1975)

    MATH  Google Scholar 

  31. Luenberger, D.: Optimization by Vector Space Methods. Wiley, New York (1969)

    MATH  Google Scholar 

  32. Källblad, S., Obloj, J., Zariphopolo, T.: Time-consistent investment under model uncertainty: the robust forward criteria. arXiv:1311.3529v2 [q-fin.PM] 14 Nov 2014

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science and Humanities, Shaqra University, Al Quwayiyah, Saudi Arabia

    Wahid Faidi

  2. University of Tunis El Manar, LAMSIN, Tunis, Tunisia

    Hanen Mezghanni

  3. University of Tunis El Manar, ENIT-LAMSIN, Tunis, Tunisia

    Mohamed Mnif

Authors
  1. Wahid Faidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hanen Mezghanni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohamed Mnif
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohamed Mnif.

Additional information

Communicated by Jean-Pierre Crouzeix.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Faidi, W., Mezghanni, H. & Mnif, M. Expected Utility Maximization Problem Under State Constraints and Model Uncertainty. J Optim Theory Appl 183, 1123–1152 (2019). https://doi.org/10.1007/s10957-019-01583-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-019-01583-y

Keywords

Mathematics Subject Classification

Search

Navigation