Chen, XinfuSaunders, DavidChadam, John2023-11-072023-11-072020https://doi.org/10.1016/j.jmaa.2019.123559http://hdl.handle.net/10012/20091The final publication is available at Elsevier via https://doi.org/10.1016/j.jmaa.2019.123559. © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We analyze the optimal withdrawal time for an investor in a hedge fund with a first-loss or shared-loss fee structure, given as the solution of an optimal stopping problem on the fund's assets with a piecewise linear payoff function. Assuming that the underlying follows a geometric Brownian motion, we present a complete solution of the problem in the infinite horizon case, showing that the continuation region is a finite interval, and that the smooth-fit condition may fail to hold at one of the endpoints. In the finite horizon case, we show the existence of a pair of optimal exercise boundaries and analyze their properties, including smoothness and convexity.enoptimal stoppingfree boundary problemsmathematical financevariational inequalitiesStefan problemArticle