Merton's Problem with Recursive Perturbed Utility

2026-02-14

AI Summary

This study addresses the inability of the classic Merton model to account for investors’ preference for randomized decisions and the difficulty of extending existing perturbed utility theories to dynamic settings. The authors propose a Recursive Perturbed Utility (RPU) framework that embeds entropy-based randomization preferences into a recursive aggregator, yielding a well-posed dynamic investment model in Markovian incomplete markets. This approach endogenously balances randomization against bequest utility over time, preventing excessive randomization. The paper derives, for the first time, a closed-form Gaussian optimal policy: its risk exposure follows a wealth-independent normal distribution, with mean composed of myopic and hedging components and variance inversely proportional to risk aversion and volatility. While randomization preferences induce only first-order deviations in the optimal policy, they result solely in higher-order welfare losses.

Technology Category

Reasoning under Uncertainty: Decision/Utility TheoryGame Theory and Economic Paradigms: Imperfect InformationKnowledge Representation and Reasoning: Preferences

Application Category

User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systemsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

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.

Problem

• Merton's problem
• randomization preference
• recursive utility
• stochastic choice
• dynamic decision-making

Innovation

• Recursive Perturbed Utility
• Stochastic Portfolio Choice
• Entropy-based Randomization
• Incomplete Markets
• Dynamic Stochastic Differential Utility