Asia-Pacific Financial Markets

, Volume 18, Issue 1, pp 89–103 | Cite as

A Note on Utility Maximization with Unbounded Random Endowment

  • Keita Owari


This paper addresses the applicability of the convex duality method for utility maximization, in the presence of random endowment. When the underlying price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true, for a wide class of utility functions and unbounded random endowments. We show this duality by exploiting Rockafellar’s theorem on integral functionals, to a random utility function.


Utility maximization Convex duality method Martingale measures 


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  1. Becherer D. (2003) Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insurance, Mathematics and Economics 33: 1–28CrossRefGoogle Scholar
  2. Bellini F., Frittelli M. (2002) On the existence of minimax martingale measures. Mathematical Finance 12: 1–21CrossRefGoogle Scholar
  3. Biagini, S., Frittelli, M., & Grasselli, M. (2010). Indifference price with general semimartingales. Mathematical Finance. (to appear).
  4. Cvitanić J., Schachermayer W., Wang H. (2001) Utility maximization in incomplete markets with random endowment. Finance and Stochastics 5: 259–272CrossRefGoogle Scholar
  5. Delbaen F., Grandits P., Rheinländer T., Samperi D., Schweizer M., Stricker C. (2002) Exponential hedging and entropic penalties. Mathematical Finance 12: 99–123CrossRefGoogle Scholar
  6. Dunford N., Schwartz J. T. (1958) Linear operators. Part I. General theory. Interscience, New YorkGoogle Scholar
  7. Frittelli, M., & Rosazza Gianin, E. (2004). Equivalent formulations of reasonable asymptotic elasticity. Technical Report 12, Department di Matematica per le Decisioni, University of Florence.Google Scholar
  8. He H., Pearson N. D. (1991) Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite-dimensional case. Journal of Economic Theory 54: 259–304CrossRefGoogle Scholar
  9. Hewitt, E., & Stromberg, K. (1975). Real and abstract analysis. A modern treatment of the theory of functions of a real variable. Graduate texts in mathematics (Vol. 25). Berlin: Springer-Verlag.Google Scholar
  10. Hugonnier J., Kramkov D. (2004) Optimal investment with random endowments in incomplete markets. The Annals of Applied Probability 14: 845–864CrossRefGoogle Scholar
  11. Hugonnier J., Kramkov D., Schachermayer W. (2004) On utility based pricing of contingent claims in incomplete markets. Mathematical Finance 15: 203–212CrossRefGoogle Scholar
  12. Jacod, J. (1979). Calcul Stochastique et Problèmes des Martingales. Lecture notes in mathematics (Vol. 714). Berlin: Springer-Verlag.Google Scholar
  13. Jacod, J. (1980). Intégrales stochastiques par rapport à une semi-martingale vectorielle et changements de filtration. In: Séminaire de Probabilités XIV, Lecture Notes in Mathematics (vol. 784, pp. 161–172). Berlin: Springer-Verlag.Google Scholar
  14. Kabanov Y. M., Stricker C. (2002) On the optimal portfolio for the exponential utility maximization: Remarks to the six-author paper. Mathematical Finance 12: 125–134CrossRefGoogle Scholar
  15. Karatzas I., Lehoczky J. P., Shreve S. E., Xu G. L. (1991) Martingale and duality methods for utility maximization in an incomplete market. SIAM Journal of Control and Optimization 29: 702–730CrossRefGoogle Scholar
  16. Kramkov D., Schachermayer W. (1999) The asymptotic elasticity of utility functions and optimal investment in incomplete markets. The Annals of Applied Probability 9: 904–950CrossRefGoogle Scholar
  17. Kramkov D., Schachermayer W. (2003) Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. The Annals of Applied Probability 13: 1504–1516CrossRefGoogle Scholar
  18. Owari, K. (2008). Robust exponential hedging and indifference valuation. Discussion Paper No. 2008-9, Hitotsubashi University.
  19. Owen M. P. (2002) Utility based optimal hedging in incomplete markets. The Annals of Applied Probability 12: 691–709CrossRefGoogle Scholar
  20. Owen M. P., Žitković G. (2009) Optimal investment with an unbounded random endowment and utility-based pricing. Mathematical Finance 19: 129–159CrossRefGoogle Scholar
  21. Rockafellar R. T. (1966) Extension of fenchel’s duality theorem for convex funcitons. Duke Mathematical Journal 33: 81–89CrossRefGoogle Scholar
  22. Rockafellar R. T. (1971) Integrals which are convex functionals. II. Pacific Journal of Mathematics 39: 439–469Google Scholar
  23. Rockafellar, R. T., & Wets, R. J. B. (1998). Variational analysis. Grundlehren der mathematischen wissenschaften (Vol. 317). Berlin: Springer-Verlag.Google Scholar
  24. Schachermayer W. (2001) Optimal investment in incomplete markets when wealth may become negative. The Annals of Applied Probability 11: 694–734CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Graduate School of Economics, Hitotsubashi UniversityTokyoJapan

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