Computational Methods and Verification Theorem for Portfolio-Consumption Optimization under Exponential O-U Dynamics

TL;DR

This paper develops a novel numerical approach for solving high-dimensional portfolio-consumption optimization problems with assets following exponential Ornstein-Uhlenbeck dynamics, providing a verification theorem and demonstrating superior accuracy and efficiency.

Contribution

It introduces a hybrid numerical method combining exponential Rosenbrock and Runge-Kutta techniques, along with a verification theorem for optimal control in complex stochastic models.

Findings

Proposed method outperforms grid-based methods in accuracy and computational efficiency.

Numerical optimal policies outperform other admissible strategies.

Verification theorem ensures the existence of optimal solutions.

Abstract

In this paper, we focus on the problem of optimal portfolio-consumption policies in a multi-asset financial market, where the n risky assets follow Exponential Ornstein-Uhlenbeck processes, along with one risk-free bond. The investor's preferences are modeled using Constant Relative Risk Aversion utility with state-dependent stochastic discounting. The problem can be formulated as a high-dimensional stochastic optimal control problem, wherein the associated value function satisfies a Hamilton-Jacobi-Bellman (HJB) equation, which constitutes a necessary condition for optimality. We apply a variable separation technique to transform the HJB equation to a system of ordinary differential equations (ODEs). Then a class of hybrid numerical approaches that integrate exponential Rosenbrock-type methods with Runge-Kutta methods is proposed to solve the ODE system. More importantly, we establish a rigorous verification theorem that provides sufficient conditions for the existence of value function and admissible optimal control, which can be verified numerically. A series of experiments are performed, demonstrating that our proposed method outperforms the conventional grid-based method in both accuracy and computational cost. Furthermore, the numerically derived optimal policy achieves superior performance over all other considered admissible policies.