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Error estimates for finite-dimensional approximations of Hamilton-Jacobi-Bellman equations on the Wasserstein space

Samuel Daudin, Joe Jackson, Benjamin Seeger

TL;DR

This work quantifies the convergence of finite-dimensional HJB equations $V^N$ to the lifted Wasserstein-HJB solution $U$ for problems with purely common noise and nonconvex Hamiltonians. It leverages Lions’ lifting to formulate a Hilbert-space (Lions) viscosity framework and implements a doubling-of-variables strategy augmented by novel simultaneous quantization techniques to control discretization errors. The authors establish dimension-dependent algebraic convergence rates, proving that in 1D the rate is favorable while higher dimensions incur degradation; they also discuss improvements under stronger structural assumptions such as bounded state spaces, periodicity, and smoother data, and they acknowledge a fundamental lower bound $N^{-1/d}$ guiding the rate. Overall, the results provide rigorous, quantitative error bounds for approximating HJB equations on measure spaces by finite-dimensional, tractable formulations, with implications for numerical methods in mean-field-type control under common noise.

Abstract

In this paper, we study a Hamilton-Jacobi-Bellman (HJB) equation set on the Wasserstein space $\mathcal{P}_2(\mathbb{R}^d)$, with a second order term arising from a purely common noise. We do not assume that the Hamiltonian is convex in the momentum variable, which means that we cannot rely on representation formulas coming from mean field control. In this setting, Gangbo, Mayorga, and Święch showed via viscosity solutions methods that the HJB equation on $\mathcal{P}_2(\mathbb{R}^d)$ can be approximated by a sequence of finite-dimensional HJB equations. Our main contribution is to quantify this convergence result. The proof involves a doubling of variables argument, which leverages the Hilbertian approach of P.L. Lions for HJB equations in the Wasserstein space, rather than working with smooth metrics which have been used to obtain similar results in the presence of idiosyncratic noise. In dimension one, our doubling of variables argument is made relatively simply by the rigid structure of one-dimensional optimal transport, but in higher dimension the argument is significantly more complicated, and relies on some estimates concerning the "simultaneous quantization" of probability measures.

Error estimates for finite-dimensional approximations of Hamilton-Jacobi-Bellman equations on the Wasserstein space

TL;DR

This work quantifies the convergence of finite-dimensional HJB equations to the lifted Wasserstein-HJB solution for problems with purely common noise and nonconvex Hamiltonians. It leverages Lions’ lifting to formulate a Hilbert-space (Lions) viscosity framework and implements a doubling-of-variables strategy augmented by novel simultaneous quantization techniques to control discretization errors. The authors establish dimension-dependent algebraic convergence rates, proving that in 1D the rate is favorable while higher dimensions incur degradation; they also discuss improvements under stronger structural assumptions such as bounded state spaces, periodicity, and smoother data, and they acknowledge a fundamental lower bound guiding the rate. Overall, the results provide rigorous, quantitative error bounds for approximating HJB equations on measure spaces by finite-dimensional, tractable formulations, with implications for numerical methods in mean-field-type control under common noise.

Abstract

In this paper, we study a Hamilton-Jacobi-Bellman (HJB) equation set on the Wasserstein space , with a second order term arising from a purely common noise. We do not assume that the Hamiltonian is convex in the momentum variable, which means that we cannot rely on representation formulas coming from mean field control. In this setting, Gangbo, Mayorga, and Święch showed via viscosity solutions methods that the HJB equation on can be approximated by a sequence of finite-dimensional HJB equations. Our main contribution is to quantify this convergence result. The proof involves a doubling of variables argument, which leverages the Hilbertian approach of P.L. Lions for HJB equations in the Wasserstein space, rather than working with smooth metrics which have been used to obtain similar results in the presence of idiosyncratic noise. In dimension one, our doubling of variables argument is made relatively simply by the rigid structure of one-dimensional optimal transport, but in higher dimension the argument is significantly more complicated, and relies on some estimates concerning the "simultaneous quantization" of probability measures.

Paper Structure

This paper contains 21 sections, 18 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Related literature
  3. Our methods
  4. Assumptions and main results
  5. Organization of the paper
  6. Preliminaries
  7. Notation
  8. $L$-viscosity solutions
  9. Lipschitz bounds
  10. Regular quantization of probability measures and random variables
  11. Error estimates
  12. The case $d = 1$
  13. The case $d \ge 2$
  14. Doubling of variables and estimates from penalization
  15. Small time estimate
  16. ...and 6 more sections

Key Result

Theorem 1.3

Let Assumption assump.main hold. Then there is a constant $C$ such that for all $N \in \mathbb N$, and all $(t,{\boldsymbol x}) \in [0,T] \times (\mathbb{R}^d)^N$, we have Moreover, if $\kappa = 0$, then the rates can be improved in dimension $d \geq 2$ as follows:

Theorems & Definitions (40)

  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Corollary 2.6
  • ...and 30 more