We study the optimal stopping problem arising from an investor determining the best time to withdraw from a hedge fund with a shared loss fee structure and a positive fee for assets under management—a decision that is critically important to the viability of such products in practice. The optimal solution is characterized as the first exit time of the fund value from a bounded region with upper and lower stopping boundaries. In the infinite horizon case, we present the complete solution to the optimal stopping problem, while in the finite horizon case we derive a pair of coupled integral equations for the stopping bounds and present an asymptotic analysis of the stopping boundaries for small time. The analysis requires new mathematical results extending techniques suitable for options with one exercise boundary to the case of a coupled pair of upper and lower boundaries.

中文翻译：

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分析从具有替代费用结构的对冲基金中撤出投资的最佳时间

我们研究了由投资者确定退出具有共同损失费用结构和对管理资产的正费用的对冲基金的最佳时间而引起的最佳停止问题——这一决定对此类产品在实践中的可行性至关重要。最优解的特征是基金价值从具有上下停止边界的有界区域的第一次退出时间。在无限水平的情况下，我们给出了最优停止问题的完整解，而在有限水平的情况下，我们推导出了一对停止边界的耦合积分方程，并对小时间停止边界进行了渐近分析。