Paper: arXiv 2506.06654

Authors: Erhan Bayraktar, Bingyan Han

Abstract

We present a continuous-time portfolio selection framework that reflects goal-based investment principles and mental accounting behavior. In this framework, an investor with multiple investment goals constructs separate portfolios, each corresponding to a specific goal, with penalties imposed on fund transfers between these goals, referred to as mental costs. By applying the stochastic Perron’s method, we demonstrate that the value function is the unique constrained viscosity solution of a Hamilton-Jacobi-Bellman equation system. Numerical analysis reveals several key features: the free boundaries exhibit complex shapes with bulges and notches; the optimal strategy for one portfolio depends on the wealth level of another; investors must diversify both among stocks and across portfolios; and they may postpone reallocating surplus from an important goal to a less important one until the former’s deadline approaches.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced mathematics including stochastic Perron’s method, constrained viscosity solutions of HJB equation systems, and free boundary analysis, indicating high complexity. However, it lacks empirical backtesting, datasets, or implementation details, focusing instead on theoretical proofs and numerical analysis without real-world data validation.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Goal-Based Portfolio Selection<br>w/ Mental Accounting"] --> B{"Methodology:<br>Stochastic Control &<br>Stochastic Perron's Method"}
  B --> C{"Mathematical Formulation:<br>HJB System &<br>Constrained Viscosity Solutions"}
  C --> D{"Computational Process:<br>Numerical Analysis of<br>Free Boundaries"}
  D --> E["Key Findings & Outcomes"]
  E --> E1["Complex Free Boundaries<br>(Bulges & Notches)"]
  E --> E2["Inter-Portfolio Dependency:<br>Strategy depends on<br>other portfolios' wealth"]
  E --> E3["Diversification Requirement:<br>Across assets & across<br>separate goal portfolios"]
  E --> E4["Delayed Reallocation:<br>Surplus transfer postponed<br>until deadline approaches"]

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