An Integral Equation in Portfolio Selection with Time-Inconsistent Preferences

TL;DR

This paper introduces a unified framework for solving a nonlinear integral equation in portfolio selection with time-inconsistent preferences, establishing existence and uniqueness of solutions under mild assumptions.

Contribution

It provides the first proof of existence and uniqueness of solutions for this class of integral equations in portfolio optimization.

Findings

Proved existence and uniqueness of solutions under mild conditions

Applicable to mean-variance and utility maximization problems

Framework requires minimal assumptions on market coefficients

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.