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Mean-field games with rough common noise: the compactification approach

Erhan Bayraktar, Xihao He, Xiang Yu, Fengyi Yuan

Abstract

We study mean-field game (MFG) problems with rough common noise where the representative state dynamics is governed by a controlled rough stochastic differential equation driven by an idiosyncratic Brownian motion and a deterministic rough path noise affecting the whole population. Within this new framework, we introduce a canonical weak formulation based on relaxed controls and rough martingale problems. We prove the existence of a pathwise mean-field equilibrium in this context by developing new technical tools for compactification to accommodate rough integration, which deviate substantially from classical compactification arguments in the literature. Finally, we discuss the relationship between the pathwise problem and the classical MFG problem with randomized Brownian common noise: conditioning yields the pathwise problem almost surely; and conversely, under a suitable causality/measurable-selection requirement, pathwise mean-field equilibria can be aggregated to produce randomized mean-field equilibria in the classical problem.

Mean-field games with rough common noise: the compactification approach

Abstract

We study mean-field game (MFG) problems with rough common noise where the representative state dynamics is governed by a controlled rough stochastic differential equation driven by an idiosyncratic Brownian motion and a deterministic rough path noise affecting the whole population. Within this new framework, we introduce a canonical weak formulation based on relaxed controls and rough martingale problems. We prove the existence of a pathwise mean-field equilibrium in this context by developing new technical tools for compactification to accommodate rough integration, which deviate substantially from classical compactification arguments in the literature. Finally, we discuss the relationship between the pathwise problem and the classical MFG problem with randomized Brownian common noise: conditioning yields the pathwise problem almost surely; and conversely, under a suitable causality/measurable-selection requirement, pathwise mean-field equilibria can be aggregated to produce randomized mean-field equilibria in the classical problem.
Paper Structure (16 sections, 22 theorems, 130 equations)

This paper contains 16 sections, 22 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. MFG with Rough Common Noise
  3. Problem formulation and assumptions
  4. Definitions and preliminaries
  5. Non-emptiness of R(omega1,lambda,mu)
  6. Existence of Pathwise MFE
  7. Step-1: compactness of R(omega1,lambda,mu)
  8. Step-2: identify the domain of $\Phi$
  9. Step-3: the upper hemicontinuity of $\Phi$
  10. Step-4: existence of fixed point
  11. Connections to MFG with Brownian Common Noise
  12. Relaxed control formulation of MFG with Brownian common noise
  13. From Brownian common noise to rough common noise
  14. From rough common noise to Brownian common noise
  15. Causal couplings between $\Lambda$ and $W$
  16. ...and 1 more sections

Key Result

Lemma 2.4

Fix indices $(\beta,\beta')\in \Pi$, $2\leq m\leq n\leq \infty$. For any $\bm{\mu}\in \mathcal{L}(\mathbf{D}^{\beta,\beta'}_\mathbf{B} L^{m,n})$, define $\tilde{\sigma}^0_t(x):= \sigma^0(t,x,\mu_t)$. Then there exists a $\tilde{\sigma}':[0,T]\to C_b^\gamma(\mathbb{R}^d;\mathbb{R}^{d\times k\times k where the "$\lesssim$" hides constants that do not depend on $(Y,Y')$.

Theorems & Definitions (63)

  • Definition 2.1: Definition 3.1 of RSDE, stochastic controlled rough path
  • Remark 1
  • Definition 2.2: Definition 3.8 of RSDE, controlled vector field
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5: Relaxed control
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 53 more