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

TL;DR

This paper develops a new framework for controlling heterogeneous mean-field stochastic differential equations with common noise, deriving novel Riccati equations and applying results to financial models like optimal trading and systemic risk.

Contribution

It introduces the first analysis of such models, deriving infinite-dimensional Riccati equations and characterizing optimal controls explicitly.

Findings

Established existence and uniqueness of solutions to the Riccati system.

Explicitly characterized optimal control strategies.

Applied framework to financial models of trading and systemic risk.

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.