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Mean-field games with unbounded controls: a weak formulation approach to global solutions

Ulrich Horst, Takashi Sato

Abstract

We establish an existence of equilibrium result for a class of non-Markovian mean-field games with unbounded control space in weak formulation. Our result is based on new existence and stability results for quadratic-growth generalized McKean-Vlasov BSDEs. Unlike earlier approaches, our approach does not require boundedness assumptions on the model parameters or time horizons and allows for running costs that are quadratic in the control variable.

Mean-field games with unbounded controls: a weak formulation approach to global solutions

Abstract

We establish an existence of equilibrium result for a class of non-Markovian mean-field games with unbounded control space in weak formulation. Our result is based on new existence and stability results for quadratic-growth generalized McKean-Vlasov BSDEs. Unlike earlier approaches, our approach does not require boundedness assumptions on the model parameters or time horizons and allows for running costs that are quadratic in the control variable.
Paper Structure (27 sections, 31 theorems, 200 equations)

This paper contains 27 sections, 31 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Organization of the paper
  4. Notation.
  5. Setup and main results
  6. Mean-field game in weak formulation
  7. Generalized McKean-Vlasov BSDEs and MFG Equilibria
  8. A sufficient maximum principle for MFGs
  9. Comparison
  10. Admissibility
  11. Existence of solutions
  12. Generalized MV-BSDEs and MV-FBSDEs
  13. Our approach
  14. Examples
  15. The solution mapping
  16. ...and 12 more sections

Key Result

Theorem 2.5

(possamai2025non) Let $(t,x,z,m) \mapsto \Lambda_t(x,z,m)$ be a measurable maximizer of the (reduced) Hamiltonian Let $\hat{\alpha} \in \bar{\mathfrak{A}}$ be an admissible control, let $\hat{\mathbb{P}} \in \mathcal{P}(\Omega)$ be a probability measure that is absolutely continuous w.r.t. $\mathbb{P}$, and let be a triple of processes that satisfy the (generalized) MV-BSDE where $\hat{\mathcal

Theorems & Definitions (46)

  • Remark 2.2
  • Definition 2.3
  • Definition 2.4: Mean-field equilibrium
  • Theorem 2.5
  • Proposition 2.7
  • Proposition 2.9
  • Remark 2.11
  • Definition 2.12
  • Theorem 2.14
  • Theorem 2.16
  • ...and 36 more