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

Filippo de Feo, Fausto Gozzi, Andrzej Święch, Lukas Wessels

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.

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

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 of finite particle systems to a function , {which} is the unique {}-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 of the so-called ``lifted'' limit optimal control problem. Under suitable additional assumptions, we show -regularity of , we prove that projects precisely onto the value functions , 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.

Paper Structure

This paper contains 37 sections, 27 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Literature: case $H=\mathbb R^d$.
  3. Literature: infinite dimensional case.
  4. Our theoretical contributions.
  5. Comparison with the state of the art.
  6. Applications.
  7. Notation
  8. The Control Problems
  9. The Finite Particle System Control Problem
  10. The Lifted Limit Control Problem
  11. Assumptions and Preliminaries
  12. Properties of the Hamiltonian H
  13. The Operators A, e^tA, B.
  14. Properties of the coefficients of the lifted problem.
  15. The HJB Equation in the Wasserstein Space
  16. ...and 22 more sections

Key Result

Lemma 4.8

Let Assumptions Assumption_f_sigma_lipschitz(ii) and Assumption_running_terminal_cost(ii) be satisfied. Then, for every $\tilde{C}>0$, there is a constant $K>0$ such that for all $n\in \mathbb{N}$, $\mu\in\mathcal{P}_2(H)$, $\mathbf{x},\mathbf{p} \in H^n$, $|\mathbf{p}|_{H^n} \leq \tilde{C}/\sqrt{n}$.

Theorems & Definitions (60)

  • Lemma 4.8
  • proof
  • Corollary 4.9
  • Lemma 4.10
  • proof
  • Proposition 4.11
  • proof
  • Lemma 4.12
  • proof
  • Lemma 4.13
  • ...and 50 more