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Linear-quadratic optimal control for non-exchangeable mean-field SDEs and applications to systemic risk

Anna de Crescenzo, Filippo de Feo, Huyên Pham

TL;DR

This work develops a linear-quadratic optimal control framework for non-exchangeable mean-field SDEs (LQ-NEMF) with graphon-type heterogeneity, and formalizes the continuum limit over $u\in[0,1]$ to obtain an infinite-dimensional Riccati system. The main novelty is the combination of a per-$u$ standard Riccati equation and a novel abstract Riccati equation on $L^2(U\times U)$, together with linear auxiliary equations, which together yield a fundamental relation decomposing the cost and a verification theorem for the optimal feedback. Existence and uniqueness are established for the full Riccati system, and the approach is demonstrated on a systemic-risk model with heterogeneous banks, where agent heterogeneity directly affects optimal risk-sharing strategies. In the homogeneous limit, the results recover classical LQ mean-field control, while the heterogeneous setting provides explicit characterization and practical insights for financial networks and similar large-population systems.

Abstract

We study the linear-quadratic control problem for a class of non-exchangeable mean-field systems, which model large populations of heterogeneous interacting agents. We explicitly characterize the optimal control in terms of a new infinite-dimensional system of Riccati equations, for which we establish existence and uniqueness. To illustrate our results, we apply this framework to a systemic risk model involving heterogeneous banks, demonstrating the impact of agent heterogeneity on optimal risk mitigation strategies.

Linear-quadratic optimal control for non-exchangeable mean-field SDEs and applications to systemic risk

TL;DR

This work develops a linear-quadratic optimal control framework for non-exchangeable mean-field SDEs (LQ-NEMF) with graphon-type heterogeneity, and formalizes the continuum limit over to obtain an infinite-dimensional Riccati system. The main novelty is the combination of a per- standard Riccati equation and a novel abstract Riccati equation on , together with linear auxiliary equations, which together yield a fundamental relation decomposing the cost and a verification theorem for the optimal feedback. Existence and uniqueness are established for the full Riccati system, and the approach is demonstrated on a systemic-risk model with heterogeneous banks, where agent heterogeneity directly affects optimal risk-sharing strategies. In the homogeneous limit, the results recover classical LQ mean-field control, while the heterogeneous setting provides explicit characterization and practical insights for financial networks and similar large-population systems.

Abstract

We study the linear-quadratic control problem for a class of non-exchangeable mean-field systems, which model large populations of heterogeneous interacting agents. We explicitly characterize the optimal control in terms of a new infinite-dimensional system of Riccati equations, for which we establish existence and uniqueness. To illustrate our results, we apply this framework to a systemic risk model involving heterogeneous banks, demonstrating the impact of agent heterogeneity on optimal risk mitigation strategies.

Paper Structure

This paper contains 23 sections, 5 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Outline of the paper.
  4. Notations.
  5. Problem formulation
  6. State equation.
  7. Functional cost.
  8. Infinite dimensional system and fundamental relation
  9. Derivation of the Riccati equations
  10. Standard Riccati.
  11. Abstract Riccati.
  12. Linear equations.
  13. Solution of the infinite dimensional differential system
  14. Standard Riccati equation
  15. Solution of the infinite dimensional Riccati equation
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $t\in[0,T]$, $\xi\in{\cal I}$, and $\alpha\in{\cal A}$. Then there exists a unique solution $X=(X^u)_u$ to the state equation dynamics, i.e., a family $X=(X^u)_u$ of stochastic processes with values in $\mathbb{R}^d$ such that the maps $u\mapsto\mathbb{E}[X^u_s]$, $u\mapsto\mathbb{E}[X^{u,i}_{s}

Theorems & Definitions (18)

  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 8 more