SIAM Journal on Financial Mathematics, Volume 11, Issue 3, Page 750-787, January 2020.
We study an optimal investment and consumption problem on infinite horizon, under the assumption that one of the investment opportunities is a fund charging high-watermark fees. The fund and the additional risky assets follow a multidimensional geometric Lévy structure. The interest rate is constant and the utility function has constant relative risk aversion. Identifying the wealth of the investor together with the distance to paying fees as the appropriate states, we obtain a two-dimensional stochastic control problem with both jumps and reflection. We derive the Hamilton--Jacobi--Bellman integro-differential equation, reduce it to one dimension, and then show it has a smooth solution. Using verification arguments the optimal strategies are obtained in feedback form. Some numerical results display the impact of the fees on the investor.

中文翻译：

---

多维跳扩散模型中高水印费的最优投资

SIAM金融数学杂志，第11卷，第3期，第750-787页，2020年1月。
我们假设投资机会之一是收取高水印费的基金，因此在无限的范围内研究了最优投资和消费问题。基金和其他风险资产遵循多维几何Lévy结构。利率是恒定的，效用函数具有恒定的相对风险规避。通过确定投资者的财富以及支付费用的距离作为适当的状态，我们获得了一个具有跳跃和反思的二维随机控制问题。我们导出了Hamilton-Jacobi-Bellman积分微分方程，将其简化为一维，然后证明它具有光滑的解。使用验证参数，以反馈形式获得最佳策略。一些数值结果显示了费用对投资者的影响。