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Optimal control of heterogeneous mean-field stochastic differential equations with common noise and applications to financial models

Filippo de Feo, Samy Mekkaoui

TL;DR

The paper addresses optimal control for heterogeneous mean-field SDEs with common noise by formulating a linear-quadratic problem over a continuum of agents indexed by $u\in I$. It develops a novel infinite-dimensional backward Riccati system on the Hilbert spaces $L^2(I;\mathbb{R}^d)$ and $L^2(I\times I;\mathbb{R}^{d\times d})$, proves existence/uniqueness, and derives an explicit feedback form for the optimal control via a fundamental relation. The authors apply the framework to two financial problems—optimal trading with heterogeneous participants and systemic risk among heterogeneous banks—yielding concrete expressions for the optimal control and the value function in terms of the Hilbert-space Riccati variables. This work extends linear-quadratic mean-field control to settings with heterogeneity and common shocks, enabling tractable analysis of coordinated strategies and systemic risk in large-scale, graphon-interacting populations with randomness at both the idiosyncratic and aggregate levels.

Abstract

Optimal control of heterogeneous mean-field stochastic differential equations with common noise has not been addressed in the literature. In this work, we initiate the study of such models. We formulate the problem within a linear-quadratic framework, a particularly important class in control theory, typically renowned for its analytical tractability and broad range of applications. We derive a novel system of backward stochastic Riccati equations on infinite-dimensional Hilbert spaces. As this system is not covered by standard theory, we establish existence and uniqueness of solutions. We explicitly characterize the optimal control in term of the solution of such system. We apply these results to solve two problems arising in mathematical finance: optimal trading with heterogeneous market participants and systemic risk in networks of heterogeneous banks.

Optimal control of heterogeneous mean-field stochastic differential equations with common noise and applications to financial models

TL;DR

The paper addresses optimal control for heterogeneous mean-field SDEs with common noise by formulating a linear-quadratic problem over a continuum of agents indexed by . It develops a novel infinite-dimensional backward Riccati system on the Hilbert spaces and , proves existence/uniqueness, and derives an explicit feedback form for the optimal control via a fundamental relation. The authors apply the framework to two financial problems—optimal trading with heterogeneous participants and systemic risk among heterogeneous banks—yielding concrete expressions for the optimal control and the value function in terms of the Hilbert-space Riccati variables. This work extends linear-quadratic mean-field control to settings with heterogeneity and common shocks, enabling tractable analysis of coordinated strategies and systemic risk in large-scale, graphon-interacting populations with randomness at both the idiosyncratic and aggregate levels.

Abstract

Optimal control of heterogeneous mean-field stochastic differential equations with common noise has not been addressed in the literature. In this work, we initiate the study of such models. We formulate the problem within a linear-quadratic framework, a particularly important class in control theory, typically renowned for its analytical tractability and broad range of applications. We derive a novel system of backward stochastic Riccati equations on infinite-dimensional Hilbert spaces. As this system is not covered by standard theory, we establish existence and uniqueness of solutions. We explicitly characterize the optimal control in term of the solution of such system. We apply these results to solve two problems arising in mathematical finance: optimal trading with heterogeneous market participants and systemic risk in networks of heterogeneous banks.

Paper Structure

This paper contains 24 sections, 6 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Literature: standard mean-field SDEs, control, and games.
  3. Literature: heterogeneous mean-field SDEs, control and games.
  4. Literature: infinite-dimensional mean-field control and games.
  5. Our theoretical contributions.
  6. Applications in mathematical finance.
  7. Outline of the paper.
  8. Notations
  9. Problem formulation
  10. State equation.
  11. Conditional cost functional.
  12. Solution of the control problem
  13. Infinite dimensional system and fundamental relation
  14. Solvability of the infinite dimensional Backward stochastic Riccati system
  15. Solvability of $K$
  16. ...and 9 more sections

Key Result

Theorem 3.5

Let $t \in [0,T]$, $\boldsymbol{\xi} \in {\cal I}_t$ and $\boldsymbol{\alpha} \in {\cal A}$. Then, there exists a unique solution ${\bf X}=(X^u)_{u \in I}$ to the state equation eq : state equations LQ common noise in the sense of Definition def : solution space S_t. Moreover, there exists a measura

Theorems & Definitions (23)

  • Definition 3.2: Admissible conditions
  • Remark 3.3
  • Definition 3.4
  • Theorem 3.5
  • Remark 3.7
  • Remark 3.8
  • Remark 4.1
  • Definition 4.2
  • Remark 4.3
  • Remark 4.4
  • ...and 13 more