Extragradient methods for mean field games of controls and mean field type FBSDEs

In this paper we present a numerical scheme to solve coupled mean field forward-backward stochastic differential equations driven by monotone vector fields. This is based on an adaptation of so called extragradient methods by characterizing solutions as zeros of monotone variational inequalities in a Hilbert space. We first introduce the procedure in the context of mean field games of controls and highlight its connection to the fictitious play. Under sufficiently strong monotonicity assumptions, we demonstrate that the sequence of approximate solutions converges exponentially fast. Then we extend the method and main results to general forward backward systems of stochastic differential equations that do not necessarily stem from optimal control.