# Robust utility maximization for complete and incomplete market models

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• Anne Gundel

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## Abstract

We investigate the problem of maximizing the robust utility functional $\inf_{Q \in \mathcal{Q}} E_Qu(X)$ . We give the dual characterization for its solution for both a complete and an incomplete market model. To this end, we introduce the new notion of reverse f-projections and use techniques developed for f-divergences. This is a suitable tool to reduce the robust problem to the classical problem of utility maximization under a certain measure: the reverse f-projection. Furthermore, we give the dual characterization for a closely related problem, the minimization of expenditures given a minimum level of expected utility in a robust setting and for an incomplete market. Copyright Springer-Verlag Berlin/Heidelberg 2005

## Suggested Citation

• Anne Gundel, 2005. "Robust utility maximization for complete and incomplete market models," Finance and Stochastics, Springer, vol. 9(2), pages 151-176, April.
• Handle: RePEc:spr:finsto:v:9:y:2005:i:2:p:151-176
DOI: 10.1007/s00780-004-0148-1
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File URL: http://hdl.handle.net/10.1007/s00780-004-0148-1

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## Citations

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Cited by:

1. Revaz Tevzadze & Teimuraz Toronjadze & Tamaz Uzunashvili, 2013. "Robust utility maximization for a diffusion market model with misspecified coefficients," Finance and Stochastics, Springer, vol. 17(3), pages 535-563, July.
2. Gundel, Anne & Weber, Stefan, 2008. "Utility maximization under a shortfall risk constraint," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1126-1151, December.
3. Daniel Bartl, 2016. "Exponential utility maximization under model uncertainty for unbounded endowments," Papers 1610.00999, arXiv.org, revised Jun 2017.
4. Alexander Schied, 2007. "Robust Optimal Control for a Consumption-investment Problem," SFB 649 Discussion Papers SFB649DP2007-026, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
5. Alexander Schied, 2008. "Robust optimal control for a consumption-investment problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(1), pages 1-20, February.
6. Constantinos Kardaras & Scott Robertson, 2010. "Robust maximization of asymptotic growth," Papers 1005.3454, arXiv.org, revised Aug 2012.
7. Daniel Bartl & Patrick Cheridito & Michael Kupper, 2017. "Robust expected utility maximization with medial limits," Papers 1712.07699, arXiv.org.
8. Grzegorz Hara'nczyk & Wojciech S{l}omczy'nski & Tomasz Zastawniak, 2007. "Relative and Discrete Utility Maximising Entropy," Papers 0709.1281, arXiv.org.
9. A. Hoseinzadeh & G. Mohtashami Borzadaran & G. Yari, 2012. "Aspects concerning entropy and utility," Theory and Decision, Springer, vol. 72(2), pages 273-285, February.
10. Andrew E. B. Lim & J. George Shanthikumar & Gah-Yi Vahn, 2012. "Robust Portfolio Choice with Learning in the Framework of Regret: Single-Period Case," Management Science, INFORMS, vol. 58(9), pages 1732-1746, September.
11. Kardaras, Constantinos & Robertson, Scott, 2012. "Robust maximization of asymptotic growth," LSE Research Online Documents on Economics 44994, London School of Economics and Political Science, LSE Library.
12. Stefan Ankirchner, 2005. "Utility duality under additional information: conditional measures versus filtration enlargements," SFB 649 Discussion Papers SFB649DP2005-029, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
13. Wiebke Wittmüß, 2006. "Robust Optimization of Consumption with Random Endowment," SFB 649 Discussion Papers SFB649DP2006-063, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
14. Oleksii Mostovyi, 2015. "Necessary and sufficient conditions in the problem of optimal investment with intermediate consumption," Finance and Stochastics, Springer, vol. 19(1), pages 135-159, January.
15. repec:spr:compst:v:67:y:2008:i:1:p:1-20 is not listed on IDEAS

### Keywords

f-divergences; utility maximization; robust utility functionals; model uncertainty; incomplete markets; duality;

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