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Comparison for semi-continuous viscosity solutions for second order PDEs on the Wasserstein space

Erhan Bayraktar, Ibrahim Ekren, Xihao He, Xin Zhang

TL;DR

This work establishes a robust comparison principle for semi-continuous viscosity solutions of second-order PDEs on the Wasserstein space, eliminating the need for Lipschitz continuity in the Fourier-Wasserstein distance and allowing direct uniqueness in $\mathcal{P}_2(\mathbb{R}^d)$. By lifting to a finite-dimensional framework and applying Ishii's lemma with carefully derived commutator estimates, the authors prove a general comparison result under a relaxed modulus condition on the extended Hamiltonian $G^e$. Two key applications are developed: (i) the value function of a stochastic control problem with partial observation is characterized as the unique viscosity solution of its HJB equation under weaker assumptions, and (ii) a prediction problem under partial monitoring yields a PDE governing limiting regret, with the comparison principle providing a bound on the limit. The technical core combines Sobolev-product estimates and commutator bounds to handle measure-derivative terms, addressing non-compactness via moment penalization and enabling convergence analysis in the Wasserstein setting. This advances viscosity theory on spaces of probability measures and supports numerical convergence analyses for McKean–Vlasov-type problems.

Abstract

In this paper, we prove a comparison result for semi-continuous viscosity solutions of a class of second-order PDEs in the Wasserstein space. This allows us to remove the Lipschitz continuity assumption with respect to the Fourier-Wasserstein distance in AriX: 2309.05040 and obtain uniqueness by directly working in the Wasserstein space. In terms of its application, we characterize the value function of a stochastic control problem with partial observation as the unique viscosity solution to its corresponding HJB equation. Additionally, we present an application to a prediction problem under partial monitoring, where we establish an upper bound on the limit of regret using our comparison principle for degenerate dynamics.

Comparison for semi-continuous viscosity solutions for second order PDEs on the Wasserstein space

TL;DR

This work establishes a robust comparison principle for semi-continuous viscosity solutions of second-order PDEs on the Wasserstein space, eliminating the need for Lipschitz continuity in the Fourier-Wasserstein distance and allowing direct uniqueness in . By lifting to a finite-dimensional framework and applying Ishii's lemma with carefully derived commutator estimates, the authors prove a general comparison result under a relaxed modulus condition on the extended Hamiltonian . Two key applications are developed: (i) the value function of a stochastic control problem with partial observation is characterized as the unique viscosity solution of its HJB equation under weaker assumptions, and (ii) a prediction problem under partial monitoring yields a PDE governing limiting regret, with the comparison principle providing a bound on the limit. The technical core combines Sobolev-product estimates and commutator bounds to handle measure-derivative terms, addressing non-compactness via moment penalization and enabling convergence analysis in the Wasserstein setting. This advances viscosity theory on spaces of probability measures and supports numerical convergence analyses for McKean–Vlasov-type problems.

Abstract

In this paper, we prove a comparison result for semi-continuous viscosity solutions of a class of second-order PDEs in the Wasserstein space. This allows us to remove the Lipschitz continuity assumption with respect to the Fourier-Wasserstein distance in AriX: 2309.05040 and obtain uniqueness by directly working in the Wasserstein space. In terms of its application, we characterize the value function of a stochastic control problem with partial observation as the unique viscosity solution to its corresponding HJB equation. Additionally, we present an application to a prediction problem under partial monitoring, where we establish an upper bound on the limit of regret using our comparison principle for degenerate dynamics.

Paper Structure

This paper contains 13 sections, 9 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Derivatives on the space of probability measures
  5. The non-linear PDE and its viscosity solution
  6. Ishii's Lemma and the Comparison principle
  7. Applications
  8. An application to controlled stochastic filtering problems
  9. An application to prediction problems under partial monitoring
  10. Proofs of the results in Section \ref{['sec:appli-1']}
  11. Estimates on the Sobolev norms of products
  12. Commutator estimates
  13. Proof of Theorem \ref{['thm:viscosity_property']}

Key Result

Lemma 3.1

Suppose that $u, -v: [0,T] \times \mathcal{P}_2({\mathbb R^d}) \rightarrow \mathbb R$ are bounded upper-semicontinuous functions. For any $\delta>0$, introduce Then there exists a local maximum $(\theta^*,\iota^*)=((t^*,\mu^*,m^*),(s^*,\nu^*,n^*))$ of Assume $\theta^*, \iota^*$ are in the interior $[0,T) \times \mathcal{P}_2(\mathbb R^d) \times \mathbb R^d$, and denote where $F_k^*(\mu), F_k^*

Theorems & Definitions (25)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 3.1: Ishii's Lemma
  • proof
  • Remark 3.3
  • Theorem 3.1: Comparison Principle
  • ...and 15 more