Path integral control under McKean-Vlasov dynamics

TL;DR

This paper explores the complex McKean-Vlasov optimal control problem, analyzing various formulations and establishing the dynamic programming principle using measure-theoretic tools in a novel framework.

Contribution

It introduces a unified measure-based framework for McKean-Vlasov control problems, enabling the application of classical methods to establish the dynamic programming principle.

Findings

Unified measure-based framework for control problems

Dynamic programming principle established under broad conditions

Applicability to financial market models with law-dependent dynamics

Abstract

We investigate the complexities of the McKean-Vlasov optimal control problem, exploring its various formulations such as the strong and weak formulations, as well as both Markovian and non-Markovian setups within financial markets. Furthermore, we examine scenarios where the law governing the control process impacts the dynamics of options. By conceptualizing controls as probability measures on a fitting canonical space with filtrations, we unlock the potential to devise classical measurable selection methods, conditioning strategies, and concatenation arguments within this innovative framework. These tools enable us to establish the dynamic programming principle under a wide range of conditions.