Coordinated Mean-Field Control for Systemic Risk

TL;DR

The work tackles systemic risk by embedding robust control within a mean-field framework that jointly optimizes a policy rate and supervisory intensity under model uncertainty. It introduces a state-dependent variance weight that couples mean and dispersion dynamics, and derives explicit feedback controls via a coupled Riccati system within a tractable LQ-MFC structure. The authors establish viscosity-solutions for the robust HJBI equation, prove a comparison-based uniqueness result, and provide a verification theorem with saddle-point structure, yielding closed-form control rules despite adversarial distortions. Simulations reveal distinct robustness-regimes and coordination phenomena between monetary and supervisory tools, emphasizing instrument complementarity and the risk of loss-of-control under strong adversaries.

Abstract

We develop a robust linear-quadratic mean-field control framework for systemic risk under model uncertainty, in which a central bank jointly optimizes interest rate policy and supervisory monitoring intensity against adversarial distortions. Our model features multiple policy instruments with interactive dynamics, implemented via a variance weight that depends on the policy rate, generating coupling effects absent in single-instrument models. We establish viscosity solutions for the associated HJB--Isaacs equation, prove uniqueness via comparison principles, and provide verification theorems. The linear-quadratic structure yields explicit feedback controls derived from a coupled Riccati system, preserving analytical tractability despite adversarial uncertainty. Simulations reveal distinct loss-of-control regimes driven by robustness-breakdown and control saturation, alongside a pronounced asymmetry in sensitivity between the mean and variance channels. These findings demonstrate the importance of instrument complementarity in systemic risk modeling and control.

Figures (6)

• Figure 1: System dynamics under different levels of adversary strength. Panels show trajectories for mean liquidity $m_t$, variance $v_t$, policy rate $u_t$, and monitoring intensity $\pi_t$. As $\lambda$ increases, $v_T$ is pushed upward, settling at a non-zero steady state in the strong-adversary case. Note in panels (a) to (c) that monitoring $\pi_t$ remains positive even after variance $v_t$ reaches zero, illustrating the over-monitoring discussed in Remarks \ref{['rema:over']} and \ref{['rema:over_scope']}.
• Figure 2: Adversary strength analysis sweeping $\lambda \in [0, 0.2]$. (a) Total cost $J$, (b) average control levels, and (c) peak adversarial distortions. Within this stable region, $J$ is primarily driven by $\lambda_v$, while no control saturation occurs (see Remarks \ref{['rema:over']} and \ref{['rema:over_scope']}).
• Figure 3: Robustness--efficiency tradeoff. (a)--(d): Heatmaps of cost, controls, and terminal variance over the $(\lambda_m, \lambda_v)$ plane. (e)--(f): Cross-sectional policy response curves fixing one adversary parameter. The heatmaps show the asymmetric policy response to $\lambda_m$ and $\lambda_v$, with $J$ increasing along $\lambda_v$ but remaining insensitive to $\lambda_m$.
• Figure 4: Parameter sensitivity analysis. $J$ and $v_T$ are most sensitive to $\chi$, $\beta$, and $R$. The increase in $J$ with $\chi$ (panel (b)) reflects the over-monitoring cost at the $v_t=0$ boundary (Remarks \ref{['rema:over']} and \ref{['rema:over_scope']}).
• Figure 5: Control saturation analysis. Increasing monitoring effectiveness $\chi$ not only saturates $\pi_t$ but also drives $u_t$ to its bound.

Theorems & Definitions (52)

• Remark 2.1: mean reversion mechanism
• Remark 2.2: model specification
• Remark 2.3: variance dynamics and common noise
• Remark 2.4: terminal variance and cost structure
• Remark 2.5: motivation for $\kappa > 0$
• Remark 2.6
• Definition 3.2: lower and upper values
• Proposition 3.3: DPP and terminal condition
• Definition 3.4: Isaacs Hamiltonian
• Proposition 3.5: HJBI for the value function