Non-Equivalence of Stochastic Optimal Control Problems with Open and Closed Loop Controls

TL;DR

This paper demonstrates that in path-dependent stochastic control problems, the value functions for open-loop and closed-loop controls can differ, contradicting previous assumptions of their equivalence under regularity conditions.

Contribution

It provides a counterexample showing the non-equivalence of value functions for open-loop and closed-loop controls in path-dependent stochastic control problems.

Findings

Open-loop and closed-loop value functions can differ in path-dependent settings.

Previous equivalence results do not hold without additional regularity.

Counterexample highlights the need for careful control formulation in stochastic problems.

Abstract

For an optimal control problem of an It\^o's type stochastic differential equation, the control process could be taken as open-loop or closed-loop forms. In the standard literature, provided appropriate regularity, the value functions under these two types of controls are equal and are the unique (viscosity) solution to the corresponding (path-dependent) HJB equation. In this short note, we provide a counterexample in the path dependent setting showing that these value functions can be different in general.