Erratum to: Utility maximization in incomplete markets with random endowment

K. Larsen, M. Soner, and G. Zitkovic kindly pointed out to us an error in our paper [1] which appeared in 2001 in this journal. They also provide an explicit counter-example in [4]. In Theorem 3.1 of [1] 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 [1] 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 the assertion on the uniqueness of the element ŷ in the supergradient of u(x) is also incorrect. Similarly, the assertion in Theorem 3.1 (ii) that ŷ and x are related via ŷ = u′(x) is incorrect. It should be replaced by the relation x = −v′(ŷ) or, equivalently, by requiring that ŷ is in the supergradient of u(x). To the best of our knowledge all the other statements in [1] are correct. As we believe that the counter-example in [4] is beautiful and instructive in its own right we take the opportunity to present it in some detail. ∗Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, [email protected]. Partially supported by the Austrian Science Fund (FWF) under grant P25815 and under grant P28861 and by the Vienna Science and Technology Fund (WWTF) under grant MA14-008.