On classical solutions to the mean field game system of controls

Abstract

We consider a class of mean field games in which the optimal strategy of a representative agent depends on the statistical distribution of the states and controls.We prove some existence results for the forward-backward system of PDEs under rather natural assumptions. The main step of the proof consists of obtaining a priori estimates on the gradient of the value function by Bernstein's method. Uniqueness is also proved under more restrictive assumptions.Finally, we discuss some examples to which the previously mentioned existence (and possibly uniqueness) results apply.

Recent Developments in Machine Learning Methods for Stochastic Control and Games

“…In the presence of interactions through the action distribution; see, e.g., [118] for the existence of classical solutions to the PDE system under suitable assumptions. MFC problems also give rise to analogous forward-backward PDE systems, except that the solution u of the backward equation is not interpreted as a value function of an optimal control problem but rather as an adjoint state.…”

Mean Field Games of Controls: Propagation of Monotonicities

“…The wellposedness of the above MFGC system has been investigated by many authors in recent years, essentially in the case β = 0 and b(x, a, ν) = a. For example, Gomes-Patrizi-Voskanyan [23], Kobeissi [28], Graber-Mayorga [26] investigated the system under some smallness conditions, and the global wellposedness (especially the uniqueness) was studied by Gomes-Voskanyan [24,25], Carmona-Lacker [17], Carmona-Delarue [15], Cardaliaguet-Lehalle [12], Kobeissi [27], under the crucial Lasry-Lions monotonicity condition. We also refer to Djete [19] for some convergence analysis from N -player games to MFGCs and Achdou-Kobeissi [2] for some numerical studies of MFGCs, without requiring the uniqueness of the equilibria.…”

On quasi-stationary Mean Field Games of Controls

“…This leads to the study a class of quasi-stationary MFG systems, where a stationary HJB equation is coupled with an evolutive Fokker-Planck equation. Recently, in [5,6,10,12,13,14], it has been introduced a significant generalization of the MFG model, called MFG of Controls, where the strategies of the agents depend not only on the position of other players but also on their strategy. The corresponding MFG system, with respect to the classical one, involves an additional fixed-point equation for the joint distribution of agent state and control.…”

“…This procedure needs the well posedness of the third equation alone which in turns is obtained solving another fixed point problem. Theorems 3.4, 4.6 and 4.10 follow this strategy which is similar to the ones in [5] and [12]. On the other hand, in Theorem 4.2 we adopt a different approach looking for a fixed point of a unique map and without solving separately the third equation in (1.1).…”