The Bellman equation for power utility maximization with semimartingales

TL;DR

This paper develops a Bellman equation framework for power utility maximization in general semimartingale models, characterizing optimal strategies via the opportunity process and providing verification theorems.

Contribution

It introduces a Bellman equation approach for utility maximization with semimartingales and describes optimal strategies through the opportunity process.

Findings

Optimal strategies solve the Bellman equation.

Opportunity process is the minimal solution of the Bellman equation.

Verification theorems confirm the optimality of strategies.

Abstract

We study utility maximization for power utility random fields with and without intermediate consumption in a general semimartingale model with closed portfolio constraints. We show that any optimal strategy leads to a solution of the corresponding Bellman equation. The optimal strategies are described pointwise in terms of the opportunity process, which is characterized as the minimal solution of the Bellman equation. We also give verification theorems for this equation.