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Control of McKean--Vlasov SDEs with Contagion Through Killing at a State-Dependent Intensity

Ben Hambly, Philipp Jettkant

TL;DR

The paper develops a rigorous framework for controlling McKean–Vlasov SDEs with contagion-through-killing and a common noise, modeling systemic risk and government intervention. It establishes convergence of nearly optimal finite-particle controls to a mean-field control problem via a controlled martingale formulation and relaxed controls, and proves equivalences between strong and smooth relaxed formulations under a linear-convex structure. It also analyzes the singular limit where killing occurs instantly upon hitting a boundary, proving existence results for MV SDEs with singular interaction and connecting the regularised model to absorption-type dynamics. Finally, it provides numerical schemes (finite elements for stochastic Fokker–Planck equations and policy-gradient optimization) and demonstrates applications to systemic-risk control in financial networks.

Abstract

We consider a novel McKean--Vlasov control problem with contagion through killing of particles and common noise. Each particle is killed at an exponential rate according to an intensity process that increases whenever the particle is located in a specific region. The removal of a particle pushes others towards the removal region, which can trigger cascades that see particles exiting the system in rapid succession. We study the control of such a system by a central agent who intends to preserve particles at minimal cost. Our theoretical contribution is twofold. Firstly, we rigorously justify the McKean--Vlasov control problem as the limit of a corresponding sequences of controlled finite particle systems. Our proof is based on a controlled martingale problem and tightness arguments. Secondly, we connect our framework with models in which particles are killed once they hit the boundary of the removal region. We show that these models appear in the limit as the exponential rate tends to infinity. As a corollary, we obtain new existence results for McKean--Vlasov SDEs with singular interaction through hitting times which extend those in the established literature. We conclude the paper with numerical investigations of our model applied to government control of systemic risk in financial systems.

Control of McKean--Vlasov SDEs with Contagion Through Killing at a State-Dependent Intensity

TL;DR

The paper develops a rigorous framework for controlling McKean–Vlasov SDEs with contagion-through-killing and a common noise, modeling systemic risk and government intervention. It establishes convergence of nearly optimal finite-particle controls to a mean-field control problem via a controlled martingale formulation and relaxed controls, and proves equivalences between strong and smooth relaxed formulations under a linear-convex structure. It also analyzes the singular limit where killing occurs instantly upon hitting a boundary, proving existence results for MV SDEs with singular interaction and connecting the regularised model to absorption-type dynamics. Finally, it provides numerical schemes (finite elements for stochastic Fokker–Planck equations and policy-gradient optimization) and demonstrates applications to systemic-risk control in financial networks.

Abstract

We consider a novel McKean--Vlasov control problem with contagion through killing of particles and common noise. Each particle is killed at an exponential rate according to an intensity process that increases whenever the particle is located in a specific region. The removal of a particle pushes others towards the removal region, which can trigger cascades that see particles exiting the system in rapid succession. We study the control of such a system by a central agent who intends to preserve particles at minimal cost. Our theoretical contribution is twofold. Firstly, we rigorously justify the McKean--Vlasov control problem as the limit of a corresponding sequences of controlled finite particle systems. Our proof is based on a controlled martingale problem and tightness arguments. Secondly, we connect our framework with models in which particles are killed once they hit the boundary of the removal region. We show that these models appear in the limit as the exponential rate tends to infinity. As a corollary, we obtain new existence results for McKean--Vlasov SDEs with singular interaction through hitting times which extend those in the established literature. We conclude the paper with numerical investigations of our model applied to government control of systemic risk in financial systems.
Paper Structure (32 sections, 33 theorems, 206 equations, 4 figures)

This paper contains 32 sections, 33 theorems, 206 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. A Model for Government Interventions in Financial Systems
  4. Main Contributions and Structure of the Paper
  5. Main Results
  6. Convergence of the Nearly Optimally Controlled Particle System
  7. The Strong Mean-Field Control Problem
  8. The Relaxed Mean-Field Control Problem
  9. Convergence to the Singular Limit
  10. Future Work
  11. Convergence of the Nearly Optimally Controlled Particle System
  12. The Finite Particle System
  13. Properties of the Limit System
  14. The Controlled Martingale Problem
  15. Equivalence between Strong and Smooth Relaxed Mean-Field Control problem
  16. ...and 17 more sections

Key Result

Theorem 2.7

Let Assumption ass:red_form_cont_prob be satisfied. Let $(\gamma^{N, 1}, \dots, \gamma^{N, N})_N$ be a sequence of controls for the particle system and denote the associated relaxed controls by $\Gamma^{N, 1}$, …, $\Gamma^{N, N}$. Define $\mu^N = \frac{1}{N}\sum_{i = 1}^N \delta_{(X^{N, i}, W^i, \Ga

Figures (4)

  • Figure 1: Convergence of stochastic gradient descent and cost for different feedback parameters $\alpha$. The dotted line in the left panel indicates the split between the two training regimes with different values for $n$, $m$, and $K$.
  • Figure 2: Heat plot of flow of subprobability distributions $\nu = (\nu_t)_{0 \leq t \leq T}$ and the control $(g^{\vartheta}(t, \cdot, \nu_t))_{0 \leq t \leq T}$ for $\alpha = 0.5$, $\alpha = 2.5$ and two different realisations of the common noise $W^0$ in the upper and lower panel.
  • Figure 3: Cost for different correlation parameters $\rho$ versus different feedback parameters $\alpha$ (note the different scales).
  • Figure 4: Heat plot of flow of subprobability distributions $\nu = (\nu_t)_{0 \leq t \leq T}$ and the control $(g^{\vartheta}(t, \cdot, \nu_t))_{0 \leq t \leq T}$ for different $\lambda_0$. The green lines in the plot on the right-hand side indicate the control boundary.

Theorems & Definitions (76)

  • Remark 2.1
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Theorem 2.9
  • Definition 2.11
  • Theorem 2.12
  • ...and 66 more