Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Intrinsic regularization by noise for $1d$ mean field games

François Delarue, Youssef Ouknine

Abstract

The purpose of this article is to show that an intrinsic noise with values in the space ${\mathcal P}({\mathbb R})$ of $1d$ probability measures may force uniqueness to first order mean field games. The structure of the noise is inspired from an earlier work [arXiv:2210.01239]. It reads as a coloured Ornstein-Uhlenbeck process with reflection on the boundary of quantile functions on the $1d$ torus, with the elements of the latter playing the role of indices for the continuum of players underpinning the game. In [arXiv:2210.01239], the semi-group generated by the noise is shown to enjoy smoothing properties that become key in the study carried out here. Although the analysis is limited to the 1d setting, this is the first example of uniqueness forcing for generic mean field games set over an infinite dimensional set of probability measures and this may be one step forward towards a more systematic regularization by noise theory for mean field games.

Intrinsic regularization by noise for $1d$ mean field games

Abstract

The purpose of this article is to show that an intrinsic noise with values in the space of probability measures may force uniqueness to first order mean field games. The structure of the noise is inspired from an earlier work [arXiv:2210.01239]. It reads as a coloured Ornstein-Uhlenbeck process with reflection on the boundary of quantile functions on the torus, with the elements of the latter playing the role of indices for the continuum of players underpinning the game. In [arXiv:2210.01239], the semi-group generated by the noise is shown to enjoy smoothing properties that become key in the study carried out here. Although the analysis is limited to the 1d setting, this is the first example of uniqueness forcing for generic mean field games set over an infinite dimensional set of probability measures and this may be one step forward towards a more systematic regularization by noise theory for mean field games.
Paper Structure (19 sections, 17 theorems, 199 equations)

This paper contains 19 sections, 17 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Mean field game with common noise and main result
  3. Candidate for being an equilibrium: reflected SDE
  4. Definition of the mean field game
  5. Characterization of mean field equilibria
  6. Analysis of the forward-backward fixed point problem
  7. Short time analysis
  8. Martingale representation
  9. Analysis over an arbitrary fixed time duration
  10. Existence and uniqueness of a mean field equilibrium (first part of Theorem \ref{['main statement']})
  11. Representation of ${\mathcal{U}}$
  12. Formula when $X$ is invertible
  13. Almost sure invertibility of the solution
  14. Main statement
  15. Connection with standard mean field games
  16. ...and 4 more sections

Key Result

Theorem 2.3

Given ${\mathcal{V}} \in {\mathcal{C}}$ and a deterministic initial condition $X_0 \in U^2({\mathds S})$, there exists a unique solution $(X_t,\eta_t)_{0 \le t \le T}$ to the rearranged SHE eq:V that satisfies Definition def:existence. Moreover, there exist constants $C,\varepsilon >0$ such that

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7
  • Proposition 2.8
  • proof
  • ...and 27 more