This paper studies a finite horizon utility maximization problem on excessive consumption under a drawdown constraint. Our control problem is an extension of the one considered in Angoshtari et al. (2019) to the model with a finite horizon and an extension of the one considered in Jeon and Oh (2022) to the model with zero interest rate. Contrary to Angoshtari et al. (2019), we encounter a parabolic nonlinear HJB variational inequality with a gradient constraint, in which some time-dependent free boundaries complicate the analysis significantly. Meanwhile, our methodology is built on technical PDE arguments, which differs from the martingale approach in Jeon and Oh (2022). Using the dual transform and considering the auxiliary variational inequality with gradient and function constraints, we establish the existence and uniqueness of the classical solution to the HJB variational inequality after the dimension reduction, and the associated free boundaries can be characterized in analytical form. Consequently, the piecewise optimal feedback controls and the time-dependent thresholds for the ratio of wealth and historical consumption peak can be obtained.

中文翻译：

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在有限范围内 drawdown constraints 下的最佳消耗


本文研究了在提款约束下过度消费的有限水平效用最大化问题。我们的控制问题是 Angoshtari 等人 （2019） 中考虑的问题扩展到有限视野的模型，以及 Jeon 和 Oh （2022） 中考虑的问题扩展到零利率模型。与 Angoshtari 等人（2019 年）相反，我们遇到了具有梯度约束的抛物线非线性 HJB 变分不等式，其中一些与时间相关的自由边界使分析变得非常复杂。同时，我们的方法建立在技术 PDE 论点之上，这与 Jeon 和 Oh （2022） 中的马丁格尔方法不同。使用对偶变换并考虑具有梯度和函数约束的辅助变分不等式，我们建立了降维后 HJB 变分不等式的经典解的存在性和唯一性，并且可以用解析形式表征相关的自由边界。因此，可以获得分段最优反馈控制和财富与历史消费峰值比率的时间相关阈值。