论文标题
在有限范围内的缩减约束下的最佳消耗
Optimal consumption under a drawdown constraint over a finite horizon
论文作者
论文摘要
本文研究了在缩减约束下过度消费的有限范围实用性最大化问题。我们的控制问题是Bahman等人中考虑的问题的扩展。 （2019年）以有限的地平线和Jeon和OH（2022）在零利率的模型中考虑的模型的扩展。与Bahman等人相反。 （2019年），我们遇到了具有梯度约束的抛物线非线性HJB变化不平等，其中某些时间依赖的自由边界显着复杂了分析。同时，我们的方法论建立在技术PDE论点上，这与Jeon和Oh（2022）的Martingale方法不同。使用双重变换并考虑具有梯度和功能约束的辅助变异不等式，我们确定了降低尺寸后HJB变化不平等的经典解决方案的存在和唯一性，并且可以以分析形式表征相关的自由边界。因此，可以获得分段最佳反馈控制和财富和历史消费峰值之比的时间依赖性阈值。
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 Bahman 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 Bahman 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.
