A continuous-time consumption-investment model with constraint is considered for a small investor whose decisions are the consumption rate and the allocation of wealth to a risk-free and a risky asset with logarithmic Brownian motion fluctuations. The consumption rate is subject to an upper bound constraint which linearly depends on the investor's wealth and bankruptcy is prohibited. The investor's objective is to maximize total expected discounted utility of consumption over an infinite trading horizon. It is shown that the value function is (second order) smooth everywhere but a unique possibility of (known) exception point and the optimal consumption-investment strategy is provided in a closed feedback form of wealth, which in contrast to the existing work does not involve the value function. According to this model, an investor should take the same optimal investment strategy as in Merton's model regardless his financial situation. By contrast, the optimal consumption strategy does depend on the investor's financial situation: he should use a similar consumption strategy as in Merton's model when he is in a bad situation, and consume as much as possible when he is in a good situation.

中文翻译：

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具有消费约束的最优消费投资模型

对于一个小投资者，考虑一个具有约束的连续时间消费投资模型，他们的决定是消费率和财富分配给具有对数布朗运动波动的无风险和风险资产。消费率受到上限约束，该上限线性取决于投资者的财富，并且禁止破产。投资者的目标是在无限的交易范围内最大化消费的总预期贴现效用。结果表明，价值函数处处（二阶）平滑，但（已知）异常点的唯一可能性和最优消费投资策略以封闭的财富反馈形式提供，与现有工作相比，涉及价值函数。根据这个模型，无论其财务状况如何，投资者都应采取与默顿模型相同的最佳投资策略。相比之下，最优消费策略确实取决于投资者的财务状况：他应该在情况不佳时使用类似于默顿模型中的消费策略，并在情况良好时尽可能多地消费。