A randomisation method for mean-field control problems with common noise

TL;DR

This paper introduces a novel randomisation approach for mean-field control problems with common noise, reformulating controls as Poisson processes and establishing a probabilistic representation via BSDEs and a dynamic programming principle.

Contribution

It develops a control randomisation framework for mean-field control with common noise, linking it to backward SDEs and a new dynamic programming principle.

Findings

Equivalent control randomisation and original problem

Representation of value function via constrained BSDE

Derivation of a randomized dynamic programming principle

Abstract

We study mean-field control (MFC) problems with common noise using the control randomisation framework, where we substitute the control process with an independent Poisson point process, controlling its intensity instead. To address the challenges posed by the mean-field interactions in this randomisation approach, we reformulate the admissible control as L 0 -valued processes adapted only to the common noise. We then construct the randomised control problem from this reformulated control process, and show its equivalence to the original MFC problem. Thanks to this equivalence, we can represent the value function as the minimal solution to a backward stochastic differential equation (BSDE) with constrained jumps. Finally, using this probabilistic representation, we derive a randomised dynamic programming principle (DPP) for the value function, expressed as a supremum over equivalent probability measures.