Paper: arXiv 2602.13544

Authors: Min Dai, Yuchao Dong, Yanwei Jia, Xun Yu Zhou

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical theoretical framework, introducing Recursive Perturbed Utility (RPU) to model randomized portfolio choices in dynamic continuous-time settings. It is pure mathematical finance theory featuring PDEs, continuous-time stochastic control, and asymptotic expansions, with no empirical backtesting or real-world market datasets evaluated.

Research Flowchart

  flowchart TD
    A[Research Goal: Incorporate Preference for Randomization into Dynamic Merton Problem] --> B[Inputs: CRRA Preferences & Markovian Incomplete Market]
    B --> C[Method: Introduce Recursive Perturbed Utility via SDU with Entropy Penalty]
    C --> D[Solve Dynamic Programming Problem & PDE for Value Function]
    D --> E[Key Outcome: Closed-Form Gaussian Optimal Portfolio Policy]
    E --> F[Findings: Mean = Myopic + Hedging Terms; Variance inversely proportional to Risk Aversion & Volatility]
    D --> G[Asymptotic Expansion: First-Order Policy Deviation vs Higher-Order Wealth Loss]