Consumption–investment problems with maximizing utility agents are usually considered from a theoretical viewpoint, aiming at closed-form solutions for the optimal policy. However, such an approach requires that the model be relatively simple: even the inclusion of nonnegativity constraints can prevent the derivation of explicit solutions. In such cases, it is necessary to solve the problem numerically, but standard dynamic programming algorithms can only solve small problems due to the curse of dimensionality. In this paper, we adapt the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve dynamic constrained consumption–investment problems with stochastic labor income numerically. Unlike classical dynamic programming approaches, SDDP allows us to analyze problems with multiple assets, and an internal sampling procedure allows the problems to have a very large, or even infinite, number of scenarios. We start with a simpler problem for which a closed-form solution is known and compare it to the optimal policy obtained by SDDP. We then illustrate the flexibility of our approach by solving a defined contribution pension fund problem with multiple assets, for which no closed-form solution is available.

中文翻译：

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通过分解算法解决约束消费投资问题


效用最大化代理的消费投资问题通常从理论角度考虑，旨在寻求最优政策的封闭式解决方案。然而，这种方法要求模型相对简单：即使包含非负约束也会阻止显式解的推导。在这种情况下，需要用数值方法解决问题，但由于维数灾难，标准动态规划算法只能解决小问题。在本文中，我们采用随机对偶动态规划（SDDP）算法来数值求解具有随机劳动收入的动态约束消费投资问题。与经典的动态规划方法不同，SDDP 允许我们分析具有多个资产的问题，并且内部采样过程允许问题具有非常大甚至无限数量的场景。我们从一个已知封闭式解决方案的简单问题开始，并将其与 SDDP 获得的最优策略进行比较。然后，我们通过解决具有多种资产的固定缴款养老基金问题来说明我们方法的灵活性，对此没有封闭式解决方案。