The Bellman equation for power utility maximization with semimartingales
Provides a theoretical foundation for utility maximization in general semimartingale settings, relevant to mathematical finance researchers.
The paper proves that optimal strategies for power utility maximization in semimartingale models with constraints satisfy the Bellman equation, and characterizes the opportunity process as its minimal solution.
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.