An Integral Equation in Portfolio Selection with Time-Inconsistent Preferences

Paper: arXiv 2412.02446

Authors: Zongxia Liang, Sheng Wang, Jianming Xia

Abstract

This paper discusses a nonlinear integral equation arising from portfolio selection with a class of time-inconsistent preferences. We propose a unified framework requiring minimal assumptions, such as right-continuity of market coefficients and square-integrability of the market price of risk. Our main contribution is proving the existence and uniqueness of the square-integrable solution for the integral equation under mild conditions. Illustrative applications include the mean-variance portfolio selection and the utility maximization with random risk aversion.

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 proof establishing the existence and uniqueness of solutions to a nonlinear integral equation in portfolio selection with time-inconsistent preferences. While highly advanced purely from a mathematical finance perspective, it is entirely theoretical with no empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Portfolio Selection with Time-Inconsistent Preferences] --> B[Inputs: Market Coefficients & Square-Integrable Market Price of Risk]
    B --> C[Methodology: Unified Framework for Nonlinear Integral Equations]
    C --> D[Mathematical Analysis: Prove Existence & Uniqueness of Square-Integrable Solutions]
    D --> E[Applications: Mean-Variance Optimization & Utility Maximization with Random Risk Aversion]
    E --> F[Key Outcome: Rigorous Equilibrium Strategies under Minimal Conditions]