  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.
