Dynamic Convex Duality in Constrained Utility Maximization

TL;DR

This paper develops a convex duality framework for constrained utility maximization, providing explicit characterizations of optimal controls and solutions via FBSDEs, simplifying complex primal problem solutions.

Contribution

It introduces a duality approach using FBSDEs to explicitly solve constrained utility maximization problems, linking primal and dual solutions dynamically.

Findings

Optimal primal wealth equals the dual adjoint process.

Explicit primal control characterized by dual FBSDEs.

Successfully solves three constrained utility maximization problems.

Abstract

In this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of FBSDEs plus additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. Moreover, we also find that the optimal primal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problems directly.