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

Charles Meynard

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.

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

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.
Paper Structure (23 sections, 21 theorems, 200 equations, 3 figures, 2 algorithms)

This paper contains 23 sections, 21 theorems, 200 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. General introduction
  3. Main contributions
  4. Organization of the paper
  5. Notation
  6. Preliminary results
  7. A primer on monotone variational inequalities in Hilbert spaces
  8. Decoupling algorithm
  9. A motivating example from mean field games
  10. Uniqueness and characterization of optimal controls
  11. An algorithm from the theory of monotone variational inequalities
  12. Link with the fictitious play
  13. Beyond mean field games: monotone FBSDE
  14. Decreasing step size
  15. Outside of the mean field regime
  16. ...and 8 more sections

Key Result

Proposition 1.2

Let $f:\left\{\right.$ be a continuous function, such that Suppose that for some $\mu\in\mathcal{P}_2(\mathbb{R}^d)$ with full support on $\mathbb{R}^d$ the following holds then the inequality is satisfied pointwise

Figures (3)

  • Figure 1: log-linear plot of the last iterate error in function of the number of iterations
  • Figure 2: $\eta(t)$
  • Figure 3: $\theta(t)$

Theorems & Definitions (51)

  • Definition 1.1
  • Proposition 1.2
  • Corollary 1.3
  • proof
  • Lemma 1.4
  • proof
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Remark 2.3
  • ...and 41 more