The Annals of Applied Probability

Utility maximization in incomplete markets

Ying Hu, Peter Imkeller, and Matthias Müller

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We consider the problem of utility maximization for small traders on incomplete financial markets. As opposed to most of the papers dealing with this subject, the investors’ trading strategies we allow underly constraints described by closed, but not necessarily convex, sets. The final wealths obtained by trading under these constraints are identified as stochastic processes which usually are supermartingales, and even martingales for particular strategies. These strategies are seen to be optimal, and the corresponding value functions determined simply by the initial values of the supermartingales. We separately treat the cases of exponential, power and logarithmic utility.

Article information

Ann. Appl. Probab., Volume 15, Number 3 (2005), 1691-1712.

First available in Project Euclid: 15 July 2005

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 91B28
Secondary: 60G44: Martingales with continuous parameter 91B70: Stochastic models 91B16: Utility theory 60H20: Stochastic integral equations 93E20: Optimal stochastic control

Financial market incomplete market maximal utility exponential utility power utility logarithmic utility supermartingale stochastic differential equation backward stochastic differential equation


Hu, Ying; Imkeller, Peter; Müller, Matthias. Utility maximization in incomplete markets. Ann. Appl. Probab. 15 (2005), no. 3, 1691--1712. doi:10.1214/105051605000000188.

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