Erratum to: Utility maximization in incomplete markets with random endowment

Jaksa Cvitanic (), Walter Schachermayer () and Hui Wang ()
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Walter Schachermayer: University of Vienna
Hui Wang: Brown University

Finance and Stochastics, 2017, vol. 21, issue 3, No 9, 867-872

Abstract: Abstract K. Larsen, M. Soner and G. Žitković kindly pointed out to us an error in our paper (Cvitanić et al. in Finance Stoch. 5:259–272, 2001) which appeared in 2001 in this journal. They also provide an explicit counterexample in Larsen et al. ( https://arxiv.org/abs/1702.02087 , 2017). In Theorem 3.1 of Cvitanić et al. (Finance Stoch. 5:259–272, 2001), it was incorrectly claimed (among several other correct assertions) that the value function u ( x ) $u(x)$ is continuously differentiable. The erroneous argument for this assertion is contained in Remark 4.2 of Cvitanić et al. (Finance Stoch. 5:259–272, 2001), where it was claimed that the dual value function v ( y ) $v(y)$ is strictly concave. As the functions u $u$ and v $v$ are mutually conjugate, the continuous differentiability of u $u$ is equivalent to the strict convexity of v $v$ . By the same token, in Remark 4.3 of Cvitanić et al. (Finance Stoch. 5:259–272, 2001), the assertion on the uniqueness of the element y ˆ $\hat{y}$ in the supergradient of u ( x ) $u(x)$ is also incorrect. Similarly, the assertion in Theorem 3.1(ii) that y ˆ $\hat{y}$ and x $x$ are related via y ˆ = u ′ ( x ) $\hat{y}=u'(x)$ is incorrect. It should be replaced by the relation x = − v ′ ( y ˆ ) $x=-v'(\hat{y})$ or, equivalently, by requiring that y ˆ $\hat{y}$ is in the supergradient of u ( x ) $u(x)$ . To the best of our knowledge, all the other statements in Cvitanić et al. (Finance Stoch. 5:259–272, 2001) are correct. As we believe that the counterexample in Larsen et al. ( https://arxiv.org/abs/1702.02087 , 2017) is beautiful and instructive in its own right, we take the opportunity to present it in some detail.

Keywords: Utility maximization; Incomplete markets; Random endowment; Duality; 91B16; 91G10; 91G20 (search for similar items in EconPapers)
JEL-codes: C61 G11 G12 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s00780-017-0331-9

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