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

Paper: arXiv 2603.05624

Authors: 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.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly rigorous theoretical analysis of non-Markovian mean-field games using generalized McKean-Vlasov BSDEs with quadratic growth. It significantly advances the mathematical literature by relaxing compactness and boundedness assumptions. However, as a purely theoretical probability paper, it lacks empirical validation or numerical implementations.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve non-Markovian MFG<br/>with unbounded controls & quadratic costs] --> B[Inputs: Weak formulation framework,<br/>unbounded control space, arbitrary time horizon]
    B --> C[Step 1: Analyze Quadratic-Growth<br/>Generalized McKean-Vlasov BSDEs]
    C --> D[Step 2: Establish new existence &<br/>stability results for MV-BSDEs]
    D --> E[Step 3: Apply Fixed-Point Argument<br/>to solve the Mean-Field Game]
    E --> F[Key Outcome: Global MFG equilibrium without<br/>boundedness assumptions on parameters]