Abstract
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.
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References
Becherer D. (2003) Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insurance, Mathematics and Economics 33: 1–28
Bellini F., Frittelli M. (2002) On the existence of minimax martingale measures. Mathematical Finance 12: 1–21
Biagini, S., Frittelli, M., & Grasselli, M. (2010). Indifference price with general semimartingales. Mathematical Finance. http://newrobin.mat.unimi.it/users/frittelli/pdf/BFGFinalRevision150920%09WEB.pdf (to appear).
Cvitanić J., Schachermayer W., Wang H. (2001) Utility maximization in incomplete markets with random endowment. Finance and Stochastics 5: 259–272
Delbaen F., Grandits P., Rheinländer T., Samperi D., Schweizer M., Stricker C. (2002) Exponential hedging and entropic penalties. Mathematical Finance 12: 99–123
Dunford N., Schwartz J. T. (1958) Linear operators. Part I. General theory. Interscience, New York
Frittelli, M., & Rosazza Gianin, E. (2004). Equivalent formulations of reasonable asymptotic elasticity. Technical Report 12, Department di Matematica per le Decisioni, University of Florence.
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: 259–304
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.
Hugonnier J., Kramkov D. (2004) Optimal investment with random endowments in incomplete markets. The Annals of Applied Probability 14: 845–864
Hugonnier J., Kramkov D., Schachermayer W. (2004) On utility based pricing of contingent claims in incomplete markets. Mathematical Finance 15: 203–212
Jacod, J. (1979). Calcul Stochastique et Problèmes des Martingales. Lecture notes in mathematics (Vol. 714). Berlin: Springer-Verlag.
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.
Kabanov Y. M., Stricker C. (2002) On the optimal portfolio for the exponential utility maximization: Remarks to the six-author paper. Mathematical Finance 12: 125–134
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: 702–730
Kramkov D., Schachermayer W. (1999) The asymptotic elasticity of utility functions and optimal investment in incomplete markets. The Annals of Applied Probability 9: 904–950
Kramkov D., Schachermayer W. (2003) Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. The Annals of Applied Probability 13: 1504–1516
Owari, K. (2008). Robust exponential hedging and indifference valuation. Discussion Paper No. 2008-9, Hitotsubashi University. http://hdl.handle.net/10086/16932.
Owen M. P. (2002) Utility based optimal hedging in incomplete markets. The Annals of Applied Probability 12: 691–709
Owen M. P., Žitković G. (2009) Optimal investment with an unbounded random endowment and utility-based pricing. Mathematical Finance 19: 129–159
Rockafellar R. T. (1966) Extension of fenchel’s duality theorem for convex funcitons. Duke Mathematical Journal 33: 81–89
Rockafellar R. T. (1971) Integrals which are convex functionals. II. Pacific Journal of Mathematics 39: 439–469
Rockafellar, R. T., & Wets, R. J. B. (1998). Variational analysis. Grundlehren der mathematischen wissenschaften (Vol. 317). Berlin: Springer-Verlag.
Schachermayer W. (2001) Optimal investment in incomplete markets when wealth may become negative. The Annals of Applied Probability 11: 694–734
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Owari, K. A Note on Utility Maximization with Unbounded Random Endowment. Asia-Pac Financ Markets 18, 89–103 (2011). https://doi.org/10.1007/s10690-010-9122-4
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DOI: https://doi.org/10.1007/s10690-010-9122-4