Expected Utility Optimization - Calculus of Variations Approach

TL;DR

This paper introduces a novel Calculus of Variations approach to derive the Hamilton-Jacobi equation for Merton's utility optimization problem, providing a new perspective beyond traditional Dynamic Programming methods.

Contribution

It is the first to apply Calculus of Variations to derive the HJ equation in stochastic control problems like Merton's problem.

Findings

CoV approach successfully derives HJ equation

Provides a new method for stochastic control problems

Potentially guarantees optimality conditions

Abstract

In this paper, I'll derive the Hamilton-Jacobi (HJ) equation for Merton's problem in Utility Optimization Theory using a Calculus of Variations (CoV) Approach. For stochastic control problems, Dynamic Programming (DP) has been used as a standard method. To the best of my knowledge, no one has used CoV for this problem. In addition, while the DP approach cannot guarantee that the optimum satisfies the HJ equation, the CoV approach does. Be aware that this is the first draft of this paper and many flaws might be introduced.