Mean-Field Games with common Poissonian noise: a Maximum Principle approach

TL;DR

This paper extends Mean-Field Games theory to include common Poissonian noise, providing a rigorous definition, equilibrium analysis, and a stochastic Pontryagin's Maximum Principle for jump-diffusions in random environments.

Contribution

It introduces a new class of Mean-Field Games with common Poissonian noise and develops a stochastic maximum principle for optimal control in this setting.

Findings

Defined Mean-Field Games with common Poissonian noise

Derived necessary optimality conditions using a stochastic maximum principle

Under convexity, established sufficiency of the conditions

Abstract

The theory of Mean-Field Games is interested in the behaviour of interacting particle systems in which the individual interaction between particles (players) decreases as the size of the population increases. In recent years, it was introduced an interesting structure for this type of games, assuming a correlated continuous source of randomness, which are called Mean-Field Games with Common Noise. In this paper, we extend this concept and provide a precise definition of a Mean-Field Game with Common Poissoninan Noise and its equilibrium. That is, a common self-exciting Poissonian structure is considered within the dynamics of the population. Then, we address the problem of optimization for jump-diffusions with random environments that lies within the definition of the MFG, and develop a stochastic version of the Pontryagin's Maximum Principle to obtain a set of necessary conditions that an optimal control must satisfy. Under additional convexity assumptions it is also shown that these conditions are also sufficient.