Error estimates of a theta-scheme for second-order mean field games

Abstract

We introduce and analyze a new finite-difference scheme, relying on the theta-method, for solving monotone second-order mean field games. These games consist of a coupled system of the Fokker–Planck and the Hamilton–Jacobi–Bellman equation. The theta-method is used for discretizing the diffusion terms: we approximate them with a convex combination of an implicit and an explicit term. On contrast, we use an explicit centered scheme for the first-order terms. Assuming that the running cost is strongly convex and regular, we first prove the monotonicity and the stability of our thetascheme, under a CFL condition. Taking advantage of the regularity of the solution of the continuous problem, we estimate the consistency error of the theta-scheme. Our main result is a convergence rate of order O ( h r ) for the theta-scheme, where ℎ is the step length of the space variable and r ∈ (0, 1) is related to the Hölder continuity of the solution of the continuous problem and some of its derivatives.