The Finite Difference Approximation Preserving Conjugate Properties of the Mean-Field Game Equations

A numerical method is presented for solving economic problems formulated in the Mean Field Game (MFG) form. The mean-field equilibrium is described by the coupled system of two parabolic partial differential equations: the Fokker-Planck-Kolmogorov equation and the Hamilton-Jacobi-Bellman one. The description is focused on the discrete approximation of these equations which accurately transfers the properties of MFG from the differential level to the discrete one. This approach results in an efficient algorithm for finding the corresponding grid control function. Contrary to other difference schemes, here the semi-Lagrangian approximation is applied which improves properties of a discrete problem. This implies the faster convergence of an iterative algorithm for the monotone minimization of the cost functional even with non-quadratic and non-symmetric contribution of control.