Optimal control of McKean-Vlasov systems under partial observation and hidden Markov switching

TL;DR

This paper develops a new approach to optimal control of McKean-Vlasov systems with hidden Markov regimes under partial observation, using a change-of-probability method to derive a Bellman equation in Wasserstein space.

Contribution

It introduces a novel change-of-probability framework to handle distribution dependence in partially observed McKean-Vlasov control problems with regime switching.

Findings

Derived a Zakai equation for the unnormalized filter.

Established a dynamic programming principle for the problem.

Formulated a Bellman equation on a Wasserstein space.

Abstract

We study a class of mean-field control problems under partial observation. The controlled dynamics are of McKean-Vlasov type and are subject to regime switching driven by a hidden Markov chain. The observation process depends on the control and on the joint distribution of the state and control, which prevents the direct application of standard filtering techniques. The main contribution of this paper is to show how this distribution dependence can be handled within a change-of-probability framework, leading to a well-posed separated control problem. We derive a Zakai equation with a specific structure for the unnormalized filter, and show that the corresponding value function satisfies a dynamic programming principle. This yields a Bellman equation posed on a convex subset of a Wasserstein space, characterizing the optimal control problem under partial observation.