Relaxed utility maximization in complete markets. (English) Zbl 1252.91076

Biagini, Sara; Guasoni, Paolo

Math. Finance 21, No. 4, 703-722 (2011).

For a relaxed investor whose relative risk aversion vanishes as wealth becomes large the utility maximization problem may not have a solution in the classical sense of the optimal payoff represented by a random variable. This was discovered by D. Kramkov and W. Schachermayer [Ann. Appl. Probab. 9, No. 3, 904–950 (1999; Zbl 0967.91017)] who introduced an asymptotic elasticity condition to exclude such situations. This paper decomposes relaxed expected utility into its classical and singular parts, represents the singular part in integral form, and proves the existence of optimal solutions for the utility maximization problem without conditions on the asymptotic elasticity.

Reviewer: Yuri Kifer (Jerusalem)

MSC:

91G20	Derivative securities (option pricing, hedging, etc.)
91G10	Portfolio theory

Keywords:

utility maximization; complete market; risk aversion; asymptotic elasticity

Citations:

Zbl 0967.91017

References:

[1]	Aliprantis, Infinite Dimensional Analysis: A Hitchhiker’s Guide (2006) · Zbl 1156.46001
[2]	Ambrosio, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich (2008) · Zbl 1210.28005
[3]	Ball, Remarks on Chacon’s Biting Lemma, Proc. Am. Math. Soc. 107 (3) pp 655– (1989) · Zbl 0678.46023
[4]	Benninga, Heterogeneity and Option Pricing, Rev. Deriv. Res. 4 (1) pp 7– (2000) · Zbl 1274.91198
[5]	Borwein, Convex Analysis and Nonlinear Optimization, Vol. 3 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC: Theory and Examples (2006) · Zbl 1116.90001
[6]	Bouchitté, Integral Representation of Convex Functionals on a Space of Measures, J. Funct. Anal. 80 (2) pp 398– (1988) · Zbl 0662.46009
[7]	Brezis, Analyse Fonctionnelle (1983)
[8]	Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Vol. 207 of Pitman Research Notes in Mathematics Series (1989)
[9]	Cvitanic , J. S. Malamud 2008 Asset Prices
[10]	Dunford, Linear Operators. Part I (1988)
[11]	Kramkov, The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets, Ann. Appl. Probab. 9 (3) pp 904– (1999) · Zbl 0967.91017
[12]	Kramkov, Necessary and Sufficient Conditions in the Problem of Optimal Investment in Incomplete Markets, Ann. Appl. Probab. 13 (4) pp 1504– (2003) · Zbl 1091.91036
[13]	Rockafellar, Conjugate Duality and Optimization, Society for Industrial and Applied Mathematics (1974) · Zbl 0296.90036
[14]	Schachermayer, Mathematical Finance-Bachelier Congress, 2000 (Paris) pp 427– (2002)
[15]	Yosida, Finitely Additive Measures, Trans. Amer. Math. Soc. 72 pp 46– (1952)