梅顿问题与递归扰动效用
Merton's Problem with Recursive Perturbed Utility
AI总结：
本文提出递归扰动效用理论，解决了动态情况下对随机化偏好问题，证明了最优投资策略在高斯分布下可闭合表达，并量化了随机化偏好的金融成本。
摘要：
经典的梅顿投资问题预测出确定性的、状态依赖性的投资组合规则；然而，实验室和实地证据表明，个体往往倾向于随机化决策，导致随机性和噪声性选择。Fudenberg等人（2015）发展了加性扰动效用理论来解释静态情况下对随机化的偏好，但这种方法在动态情况下变得不恰当或难以处理。我们引入了递归扰动效用（RPU），一种特殊的随机微分效用，它将基于熵的对随机化的偏好纳入递归聚合器中。RPU内生化了跨时期之间随机化带来的效用与遗产之间的权衡，通过一个依赖于过去累积随机化的折扣项，从而避免过度随机化并产生一个恰当的问题。在一般的马尔可夫不完全市场中，我们证明了RPU最优的投资组合策略（以风险敞露比例表示）是高斯分布的，并且可以以闭合形式表达，与财富无关。其方差与风险厌恶和股票波动性成反比，而其均值基于一个偏微分方程的解。此外，均值是短视项和跨时期对冲项（针对市场不完全性）的和，这两者交织在一起与政策随机化相关。最后，我们通过扰动效用权重的渐近展开来展示，最优均值策略在一级上偏离经典梅顿策略，而相关的相对财富损失则为更高阶，量化了对随机化的偏好的金融成本。
英文摘要：
The classical Merton investment problem predicts deterministic, state-dependent portfolio rules; however, laboratory and field evidence suggests that individuals often prefer randomized decisions leading to stochastic and noisy choices. Fudenberg et al. (2015) develop the additive perturbed utility theory to explain the preference for randomization in the static setting, which, however, becomes ill-posed or intractable in the dynamic setting. We introduce the recursive perturbed utility (RPU), a special stochastic differential utility that incorporates an entropy-based preference for randomization into a recursive aggregator. RPU endogenizes the intertemporal trade-off between utilities from randomization and bequest via a discounting term dependent on past accumulated randomization, thereby avoiding excessive randomization and yielding a well-posed problem. In a general Markovian incomplete market with CRRA preferences, we prove that the RPU-optimal portfolio policy (in terms of the risk exposure ratio) is Gaussian and can be expressed in closed form, independent of wealth. Its variance is inversely proportional to risk aversion and stock volatility, while its mean is based on the solution to a partial differential equation. Moreover, the mean is the sum of a myopic term and an intertemporal hedging term (against market incompleteness) that intertwines with policy randomization. Finally, we carry out an asymptotic expansion in terms of the perturbed utility weight to show that the optimal mean policy deviates from the classical Merton policy at first order, while the associated relative wealth loss is of a higher order, quantifying the financial cost of the preference for randomization.
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