Regularity properties in a state-constrained expected utility maximization problem

TL;DR

This paper studies a stochastic control problem involving market impact and finite fuel constraints, establishing regularity and uniqueness of solutions to facilitate further analysis of the value function.

Contribution

It proves the regularity, continuity, and uniqueness of optimal strategies in a market impact model with finite fuel constraints, advancing theoretical understanding.

Findings

Existence and uniqueness of optimal strategies under mild assumptions

Regularity and continuity of the value function

Validation of the dynamic programming principle without classical measurable selection

Abstract

We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the initial condition fulfilled by the corresponding value function and show its first regularity property. Moreover, we can prove the existence and uniqueness of optimal strategies under rather mild model assumptions. On the one hand, this result is of independent interest. On the other hand, it will then allow us to derive further regularity properties of the corresponding value function, in particular its continuity and partial differentiability. As a consequence of the continuity of the value function, we will prove the dynamic programming principle without appealing to the classical measurable selection arguments.