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The mean field game system is a system of Bellman and Kolmogorov equations. For the finite state case they are ordinary differential equations. We examine the finite state mean filed game assuming only continuity of the Kolmogorov matrix and award functions. Thus, it may turns out that the optimal feedback depends on the solution of Bellman equation discontinuously. We show that under these, rather general, assumptions the finite state mean field game can be regarded as a control problem with boundary constraints. Based on this, we describe the set of solutions of the mean field game for the given initial distribution of players as a viable set under certain dynamical system.

Bio: ‪Yurii Averboukh is a research staff member at Krasovskii Institute of Mathematics and Mechanics (Yekaterinburg, Russia) and a researcher in the Labarotory of stochastic analysis at Higher School of Economics (Moscow). He obtained the PhD degree from Institute of mathematics and mechanics of Ural Branch of Russian Academy of Sciences in 2007. Yurii’s interests lies in the field of optimal control theory and are focused on differential games, mean field games and the theory of mean field type control problems