Linear Quadratic Extended Mean Field Games and Control Problems

We provide a thorough study of a general class of linear-quadratic extended mean field games and control problems in any dimensions where the mean field terms are allowed to be unbounded and there is also the presence of cross terms in the objective functionals. Our investigation focuses on the existence of a unique equilibrium strategy for the extended mean field problems by employing the stochastic maximum principle approach and the appropriate fixed point argument. Notably, the linearity of the state and adjoint processes allows us to derive a forward-backward ordinary differential equation governing the optimal mean field term. We provide two distinct proofs, accompanied by two sufficient conditions, that establish the existence of unique equilibrium strategy over a global time horizon. Both conditions emphasize the importance of sufficiently small coefficients of sensitivity for the cross term, of state and control, and mean field term. To determine the required magnitudes of these coefficients, we utilize the singular values of appropriate matrices and Weyl’s inequalities. The proposed theory is consistent with the classical one, namely, our theoretical framework encompasses classical linear-quadratic stochastic control problems as particular cases. Additionally, we establish sufficient conditions for the existence of a unique solution to a particular class of non-symmetric Riccati equations, and we illustrate a counterexample to the existence of equilibrium strategies. Furthermore, we also apply the stochastic maximum principle approach to examine linear-quadratic extended mean field type stochastic control problems. Compared to the game setting, the existence of a unique solution to the corresponding adjoint equations of the extended mean field type control problems can always be warranted without the requirement of any sufficiently small coefficients; indeed, it can be established under a natural convexity condition. Finally, we conduct a comparative analysis between our method and the alternative master equation approach, specifically addressing the efficacy of the proposed approach in solving common practical problems, for which the explicit forms of the equilibrium strategies can be obtained directly, even over any global time horizon.