Mean Field Control with Poissonian Common Noise: A Pathwise Compactification Approach
Lijun Bo, Jingfei Wang, Xiaoli Wei, Xiang Yu
TL;DR
The paper develops a novel pathwise compactification method for mean-field control and mean-field games with Poissonian common noise by freezing a sample path of the Poisson noise (finite intensity ensures finitely many jumps). Step-1 establishes the existence of optimal pathwise relaxed controls in a noise-free auxiliary model using Skorokhod topology, yielding a measurable selection. Step-2 proves an aggregation principle via a pathwise measure-valued control and a superposition result, linking the pathwise problem back to the original model and ensuring optimality in the presence of Poissonian common noise; the framework also extends to strong mean-field equilibria in MFG. The approach preserves adaptivity to the common-noise filtration and avoids discretization of the noise, offering robust existence results under finite-intensity Poisson noise and a rigorous equivalence between formulations. Overall, the work provides a rigorous, pathwise route to existence of optimal controls and strong MFE in MFC/MFG with jump-type common shocks and highlights the role of deterministic jumping times in the concatenation-based superposition principle.
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.