Error estimates of a theta-scheme for second-order mean field games. (English) Zbl 1522.65137
Summary: 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 \(\mathcal{O}(h^r)\) for the theta-scheme, where \(h\) is the step length of the space variable and \(r \in (0, 1)\) is related to the Hölder continuity of the solution of the continuous problem and some of its derivatives.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 91-08 | Computational methods for problems pertaining to game theory, economics, and finance |
| 91A16 | Mean field games (aspects of game theory) |
| 49N80 | Mean field games and control |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
| 35Q84 | Fokker-Planck equations |
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