  In this article we prove the existence of classical solutions to a system of
mean field games arising in the study of exhaustible resource production under
market competition. Individual trajectories are modeled by a controlled
diffusion process with jumps, which adds a nonlocal term to the PDE system. The
assumptions on the Hamiltonian are sufficiently general to cover a large class
of examples proposed in the literature on Bertrand and Cournot mean field
games. Uniqueness also holds under a sufficient restriction on the structure of
the Hamiltonian, which in practice amounts to a small upper bound on the
substitutability of goods.
