The Annals of Applied Probability

Dual formulation of the utility maximization problem: The case of nonsmooth utility

B. Bouchard, N. Touzi, and A. Zeghal

Full-text: Open access

Abstract

We study the dual formulation of the utility maximization problem in incomplete markets when the utility function is finitely valued on the whole real line. We extend the existing results in this literature in two directions. First, we allow for nonsmooth utility functions, so as to include the shortfall minimization problems in our framework. Second, we allow for the presence of some given liability or a random endowment. In particular, these results provide a dual formulation of the utility indifference valuation rule.

Article information

Source
Ann. Appl. Probab., Volume 14, Number 2 (2004), 678-717.

Dates
First available in Project Euclid: 23 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1082737107

Digital Object Identifier
doi:10.1214/105051604000000062

Mathematical Reviews number (MathSciNet)
MR2052898

Zentralblatt MATH identifier
1126.91018

Subjects
Primary: 90A09 93E20: Optimal stochastic control 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56]
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 90A16

Keywords
Utility maximization incomplete markets convex duality

Citation

Bouchard, B.; Touzi, N.; Zeghal, A. Dual formulation of the utility maximization problem: The case of nonsmooth utility. Ann. Appl. Probab. 14 (2004), no. 2, 678--717. doi:10.1214/105051604000000062. https://projecteuclid.org/euclid.aoap/1082737107


Export citation

References

  • Bouchard, B. (2002). Utility maximization on the real line under proportional transaction costs. Finance and Stochastics 6 495--516.
  • Cvitanić, J. (1999). Minimizing expected loss of hedging in incomplete and constrained markets. SIAM J. Control Optim. 38 1050--1066.
  • Cvitanić, J. and Karatzas, I. (1999). On dynamic measures of risk. Finance and Stochastics 3 451--482.
  • Cvitanić, J., Schachermayer, W. and Wang, H. (2001). Utility maximization in incomplete markets with random endowment. Finance and Stochastics 5 259--272.
  • Deelstra, G., Pham, H. and Touzi, N. (2001). Dual formulation of the utility maximization problem under transaction costs. Ann. Appl. Probab. 11 1353--1383.
  • Delbaen, F. and Shachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215--250.
  • Delbaen, F., Grandits, P., Reinländer, T., Samperi, D., Schweizer, M. and Stricker, C. (2002). Exponential hedging and entropic penalties. Math. Finance 12 99--123.
  • Föllmer, H. and Leukert, P. (2000). Efficient hedging: Cost versus shortfall risk. Finance and Stochastics 4 117--146.
  • Hodges, S. and Neuberger, A. (1989). Optimal replication of contingent claims under transaction costs. Review of Futures Markets 8 222--239.
  • Hugonnier, J. and Kramkov, D. (2001). Optimal investment with a random endowment in incomplete markets. Unpublished manuscript.
  • Kabanov, Yu. and Stricker, C. (2002). On the optimal portfolio for the exponential utility maximization: Remarks to the six-author paper. Math. Finance 12 125--134.
  • Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904--950.
  • Kramkov, D. and Schachermayer, W. (2001). Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Unpublished manuscript.
  • Owen, M. (2002). Utility based optimal hedging in incomplete markets. Ann. Appl. Probab. 12 691--709.
  • Pham, H. (2000). Minimizing shortfall risk and applications to finance and insurance problems. Ann. Appl. Probab. 12 143--172.
  • Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.
  • Rogers, L. C. G. (2001). Duality in constrained optimal investment and consumption problems: A synthesis. Lectures presented at the Workshop on Financial Mathematics and Econometrics, Montreal.
  • Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Probab. 11 694--734.
  • Schachermayer, W. (2001). How potential investments may change the optimal portfolio for the exponential utility. Unpublished manuscript.