Dual formulation of the utility maximization problem: the case of nonsmooth utility. (English) Zbl 1126.91018

Summary: 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 [see e.g. M. P. Owen, Ann. Appl. Probab. 12, No. 2, 691–709 (2002; Zbl 1049.91082)] 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.


91B26 Auctions, bargaining, bidding and selling, and other market models
91B28 Finance etc. (MSC2000)
49J52 Nonsmooth analysis
49N15 Duality theory (optimization)
60H30 Applications of stochastic analysis (to PDEs, etc.)
91B62 Economic growth models
93E20 Optimal stochastic control
91B16 Utility theory


Zbl 1049.91082
Full Text: DOI arXiv Euclid


[1] Bouchard, B. (2002). Utility maximization on the real line under proportional transaction costs. Finance and Stochastics 6 495–516. · Zbl 1039.91019
[2] Cvitanić, J. (1999). Minimizing expected loss of hedging in incomplete and constrained markets. SIAM J. Control Optim. 38 1050–1066. · Zbl 1034.91037
[3] Cvitanić, J. and Karatzas, I. (1999). On dynamic measures of risk. Finance and Stochastics 3 451–482. · Zbl 0982.91030
[4] Cvitanić, J., Schachermayer, W. and Wang, H. (2001). Utility maximization in incomplete markets with random endowment. Finance and Stochastics 5 259–272. · Zbl 0993.91018
[5] Deelstra, G., Pham, H. and Touzi, N. (2001). Dual formulation of the utility maximization problem under transaction costs. Ann. Appl. Probab. 11 1353–1383. · Zbl 1012.60059
[6] Delbaen, F. and Shachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215–250. · Zbl 0917.60048
[7] Delbaen, F., Grandits, P., Reinländer, T., Samperi, D., Schweizer, M. and Stricker, C. (2002). Exponential hedging and entropic penalties. Math. Finance 12 99–123. · Zbl 1072.91019
[8] Föllmer, H. and Leukert, P. (2000). Efficient hedging: Cost versus shortfall risk. Finance and Stochastics 4 117–146. · Zbl 0956.60074
[9] Hodges, S. and Neuberger, A. (1989). Optimal replication of contingent claims under transaction costs. Review of Futures Markets 8 222–239.
[10] Hugonnier, J. and Kramkov, D. (2001). Optimal investment with a random endowment in incomplete markets. Unpublished manuscript. · Zbl 1086.91030
[11] 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. · Zbl 1073.91034
[12] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904–950. · Zbl 0967.91017
[13] Kramkov, D. and Schachermayer, W. (2001). Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Unpublished manuscript. · Zbl 1091.91036
[14] Owen, M. (2002). Utility based optimal hedging in incomplete markets. Ann. Appl. Probab. 12 691–709. · Zbl 1049.91082
[15] Pham, H. (2000). Minimizing shortfall risk and applications to finance and insurance problems. Ann. Appl. Probab. 12 143–172. · Zbl 1015.93071
[16] Rockafellar, R. T. (1970). Convex Analysis . Princeton Univ. Press. · Zbl 0193.18401
[17] 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. · Zbl 1074.91020
[18] Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Probab. 11 694–734. · Zbl 1049.91085
[19] Schachermayer, W. (2001). How potential investments may change the optimal portfolio for the exponential utility. Unpublished manuscript.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.