Optimal Consumption and Portfolio Choice with No-Borrowing Constraint in the Kim-Omberg Model

2026-03-03

AI Summary

This study addresses the optimal consumption and portfolio choice problem for an investor facing stochastic excess returns modeled by an Ornstein-Uhlenbeck process and a no-borrowing constraint. Building upon the Kim–Omberg framework, the paper introduces the borrowing prohibition for the first time and employs Lagrangian duality to transform the primal problem into a singular control problem in the dual space. The solution is characterized via a two-dimensional optimal stopping problem, which yields an analytical representation of the optimal policy. The analysis reveals significant economic effects of the no-borrowing constraint on investment behavior and extends the applicability of classical models to more realistic market settings with financial constraints.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

In this paper, we study an intertemporal utility maximization problem in which an investor chooses consumption and portfolio strategies in the presence of a stochastic factor and a no-borrowing constraint. In the spirit of the Kim-Omberg model, the stochastic factor represents the excess return of the risky asset and follows an Ornstein-Uhlenbeck process, capturing the mean reversion of expected excess returns-a feature well supported by empirical evidence in financial markets. The investor seeks to maximize expected utility from consumption, subject to the constraint that wealth remains nonnegative at all times. To address the dynamic no-borrowing constraint, we use Lagrange duality to transform the primal problem into a singular control problem in the dual space. We then characterize the solution to the dual singular control problem via an auxiliary two-dimensional optimal stopping problem featuring stochastic volatility, and subsequently retrieve the primal value function as well as the optimal portfolio and consumption plans. Finally, a numerical study is conducted to derive economic and financial implications.

Problem

• no-borrowing constraint
• optimal consumption
• portfolio choice
• stochastic factor
• intertemporal utility maximization

Innovation

• no-borrowing constraint
• Lagrange duality
• singular control
• optimal stopping
• stochastic factor