Stochastic Optimal Control of Interacting Particle Systems in Hilbert Spaces and Applications

TL;DR

This paper develops a theoretical framework for the optimal control of large systems of interacting stochastic particles in Hilbert spaces, establishing convergence results and regularity properties, with applications in economics.

Contribution

It introduces the first comprehensive limiting theory for stochastic control of interacting particles in infinite-dimensional spaces, including convergence, regularity, and control correspondence results.

Findings

Proves convergence of finite particle value functions to a mean-field limit.

Establishes $C^{1,1}$ regularity of the limit value function.

Demonstrates applications to economic models with delay and PDE-based particles.

Abstract

Optimal control of interacting particles governed by stochastic evolution equations in Hilbert spaces is an open area of research. Such systems naturally arise in formulations where each particle is modeled by stochastic partial differential equations, path-dependent stochastic differential equations (such as stochastic delay differential equations or stochastic Volterra integral equations), or partially observed stochastic systems. The purpose of this manuscript is to build the foundations for a limiting theory as the number of particles tends to infinity. We prove the convergence of the value functions $u_n$ of finite particle systems to a function $\mathcal{V}$, {which} is the unique {$L$}-viscosity solution of the corresponding mean-field Hamilton-Jacobi-Bellman equation {in the space of probability measures}, and we identify its lift with the value function $U$ of the so-called ``lifted'' limit optimal control problem. Under suitable additional assumptions, we show $C^{1,1}$-regularity of $U$, we prove that $\mathcal{V}$ projects precisely onto the value functions $u_n$, and that optimal (resp. optimal feedback) controls of the particle system correspond to optimal (resp. optimal feedback) controls of the lifted control problem started at the corresponding initial condition. To the best of our knowledge, these are the first results of this kind for stochastic optimal control problems for interacting particle systems of stochastic evolution equations in Hilbert spaces. We apply the developed theory to problems arising in economics where the particles are modeled by stochastic delay differential equations and stochastic partial differential equations.