Dynamic Lagrange Multipliers in a Non-concave Utility Framework

TL;DR

This paper introduces dynamic Lagrange multipliers to unify martingale duality and dynamic programming in non-concave utility portfolio optimization, providing new insights and numerical validation.

Contribution

It proposes a novel dynamic multiplier framework that links two classical approaches and offers economic interpretations in non-concave utility settings.

Findings

Lagrangian multiplier equals the conjugate dual point of the value function.

Dynamic multiplier process is homogeneous and relates to wealth and pricing kernel.

Classical optimal results are recovered and validated numerically.

Abstract

In continuous-time portfolio selection for non-concave utility functions, the martingale duality approach is widely adopted in complete markets, while the dynamic programming approach may sometimes lead to singular solutions of the Hamilton-Jacobi-Bellman (HJB) equation. We propose "dynamic Lagrange multipliers" in a non-concave utility framework, bridging two approaches and demonstrating that the Lagrangian multiplier function (in the martingale duality approach) equals the conjugate dual point related to the value function (in dynamic programming), which is exactly its partial derivative with respect to wealth. Moreover, the dynamic multiplier process exhibits homogeneity via the optimal wealth and pricing kernel processes, offering intuitive economic interpretations as a dynamic shadow price of the envelope theorem. Finally, classical optimal results are recovered and numerically validated by non-concave utility examples.