Mean Field Control with Poissonian Common Noise: A Pathwise Compactification Approach

TL;DR

This paper introduces a pathwise compactification method for mean-field control problems with Poissonian common noise, overcoming compactness issues by leveraging point process representations and aggregation techniques.

Contribution

It develops a novel two-step pathwise approach to establish existence and optimality of controls in mean-field problems with Poisson noise, including mean-field games.

Findings

Existence of optimal relaxed controls in the pathwise formulation.

Aggregation of solutions yields optimal controls in the original model.

Extension to mean-field games confirms the existence of strong equilibria.

Abstract

This paper contributes to the compactification approach to study mean-field control problems with Poissonian common noise. To overcome the lack of compactness and continuity issues caused by common noise, we exploit the point process representation of the Poisson random measure with finite intensity and propose a pathwise formulation in a two-step procedure by freezing a sample path of the common noise. In the first step, we establish the existence of the optimal relaxed control in the pathwise formulation as if common noise is absent, but with finite deterministic jumping times. The second step plays the key role in our approach, which is to aggregate the optimal solutions in the pathwise formulation over all sample paths of common noise and show that it yields an optimal solution in the original model. To this end, with the help of concatenation techniques, we first develop a pathwise superposition principle in the model with deterministic jumping times, drawing a relationship between the pathwise relaxed control problem and the pathwise measure-valued control problem. As a result, we can further bridge the equivalence among different problem formulations and verify that the constructed solution under aggregation is indeed optimal in the original problem. We also extend the methodology to solve mean-field games with Poissonian common noise, confirming the existence of a strong mean field equilibrium.