We develop a method to find approximate solutions, and their accuracy, to consumption–investment problems with isoelastic preferences and infinite horizon, in incomplete markets where state variables follow a multivariate diffusion. We construct upper and lower contractions; these are fictitious complete markets in which state variables are fully hedgeable, but their dynamics is distorted. Such contractions yield pointwise upper and lower bounds for both the value function and the optimal consumption of the original incomplete market, and their optimal policies are explicit in typical models. Approximate consumption–investment policies coincide with the optimal one if the market is complete or utility is logarithmic.

中文翻译：

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不完全市场中的消费

我们开发了一种方法，可以在状态变量遵循多元扩散的不完整市场中，找到具有等弹性偏好和无限期的消费-投资问题的近似解及其准确性。我们构造上下收缩；这些是虚拟的完整市场，其中状态变量可以完全套期保值，但其动态却被扭曲了。这样的收缩产生了价值函数和原始不完全市场的最优消费的逐点上限和下限，并且它们的最优策略在典型模型中是明确的。如果市场完整或效用为对数，则近似的消费－投资政策与最优政策一致。