Erratum to: Utility maximization in incomplete markets with random endowment

Abstract

K. Larsen, M. Soner and G. Žitkovic kindly pointed out to us an error in our paper (Cvitanic 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 Cvitanic et al. (Finance Stoch. 5:259–272, 2001), it was incorrectly claimed (among several other correct assertions) that the value function $u(x)$ is continuously differentiable. The erroneous argument for this assertion is contained in Remark 4.2 of Cvitanic et al. (Finance Stoch. 5:259–272, 2001), where it was claimed that the dual value function $v(y)$ is strictly concave. As the functions $u$ and $v$ are mutually conjugate, the continuous differentiability of $u$ is equivalent to the strict convexity of $v$ . By the same token, in Remark 4.3 of Cvitanic et al. (Finance Stoch. 5:259–272, 2001), the assertion on the uniqueness of the element $\hat{y}$ in the supergradient of $u(x)$ is also incorrect. Similarly, the assertion in Theorem 3.1(ii) that $\hat{y}$ and $x$ are related via $\hat{y}=u'(x)$ is incorrect. It should be replaced by the relation $x=-v'(\hat{y})$ or, equivalently, by requiring that $\hat{y}$ is in the supergradient of $u(x)$ . To the best of our knowledge, all the other statements in Cvitanic 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.