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

TL;DR

This paper introduces an extragradient numerical scheme for solving coupled mean field forward-backward stochastic differential equations, demonstrating exponential convergence under strong monotonicity assumptions and extending to general systems.

Contribution

It adapts extragradient methods to mean field games and FBSDEs, providing a convergent numerical scheme with broad applicability.

Findings

Converges exponentially fast under strong monotonicity.

Applicable to mean field games of controls and general FBSDEs.

Connects extragradient methods with fictitious play in mean field contexts.

Abstract

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.