In this study, we generalize the results of Arun (The Merton problem with a drawdown constraint on consumption. Working paper, 2013) on the optimal consumption and investment problem of an infinitely lived agent who does not accept her consumption falling below a fixed proportion of her historically highest level, the so-called drawdown constraint on consumption. We extend the results to a general class of utility functions. We use the martingale method to study the dual problem, which involves the choice of a maximum consumption process. The dual problem can be formulated as a two-dimensional singular control problem, with the free boundary depending on a state variable of the maximum process. We establish the duality theorem and provide semi-closed form solutions regarding the optimal strategies. To highlight our methodology, we present some special cases of utility functions that do not allow for the dimension reduction considered in Arun (2013).

在这项研究中，我们推广了Arun（对消费具有缩水约束的默顿问题。工作论文，2013年）关于不接受其消费量低于固定比例的无限寿命代理商的最优消费和投资问题的结果。她的历史最高水平，即所谓的消费限制。我们将结果扩展到通用类的效用函数。我们使用the方法研究双重问题，其中涉及最大消耗过程的选择。对偶问题可以表示为二维奇异控制问题，其自由边界取决于最大过程的状态变量。我们建立对偶定理，并提供关于最佳策略的半封闭形式解。为了突出我们的方法，