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Viscosity Solutions of Fully second-order HJB Equations in the Wasserstein Space

Erhan Bayraktar, Hang Cheung, Ibrahim Ekren, Jinniao Qiu, Ho Man Tai, Xin Zhang

Abstract

In this paper, we show that the value functions of mean field control problems with common noise are the unique viscosity solutions to fully second-order Hamilton-Jacobi-Bellman equations, in a Crandall-Lions-like framework. We allow the second-order derivative in measure to be state-dependent and thus infinite-dimensional, rather than derived from a finite-dimensional operator, hence the term ''fully''. Our argument leverages the construction of smooth approximations from particle systems developed by Cosso, Gozzi, Kharroubi, Pham, and Rosestolato [Trans. Amer. Math. Soc., 2023], and the compactness argument via penalization of measure moments in Soner and Yan [Appl. Math. Optim., 2024]. Our work addresses unbounded dynamics and state-dependent common noise volatility, and to our knowledge, this is the first result of its kind in the literature.

Viscosity Solutions of Fully second-order HJB Equations in the Wasserstein Space

Abstract

In this paper, we show that the value functions of mean field control problems with common noise are the unique viscosity solutions to fully second-order Hamilton-Jacobi-Bellman equations, in a Crandall-Lions-like framework. We allow the second-order derivative in measure to be state-dependent and thus infinite-dimensional, rather than derived from a finite-dimensional operator, hence the term ''fully''. Our argument leverages the construction of smooth approximations from particle systems developed by Cosso, Gozzi, Kharroubi, Pham, and Rosestolato [Trans. Amer. Math. Soc., 2023], and the compactness argument via penalization of measure moments in Soner and Yan [Appl. Math. Optim., 2024]. Our work addresses unbounded dynamics and state-dependent common noise volatility, and to our knowledge, this is the first result of its kind in the literature.
Paper Structure (13 sections, 14 theorems, 116 equations)

This paper contains 13 sections, 14 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compactness and Differentiation in the Wasserstein space
  4. Problem Formulation
  5. Associated Mean Field Control Problems
  6. Smooth Finite-dimensional Approximations of Value Function
  7. Approximation by Non-degenerate Problem
  8. Smooth Finite-dimensional Approximation of Coefficient Functions
  9. Approximation of Value Function
  10. Viscosity Solution Theory
  11. Existence
  12. Uniqueness
  13. Technical Proofs in Section \ref{['approximation']}

Key Result

Lemma 2.1

Let $K>0$ and $p_2 > p_1 \geq 1$. The set $V^{p_1,p_2}_K:=\{ \mu\in\mathcal{P}_{p_1}(\mathbb{R}^d) : M_{p_2}(\mu) \leq K\}=\{ \mu\in\mathcal{P}_{p_2}(\mathbb{R}^d) : M_{p_2}(\mu) \leq K\}$ is compact in $(\mathcal{P}_{p}(\mathbb{R}^d),\mathcal{W}_{p_1})$ for any $p\in[p_1,p_2]$.

Theorems & Definitions (34)

  • Lemma 2.1
  • proof
  • Remark 2.1
  • Definition 2.2
  • Remark 2.2
  • Definition 2.3
  • Remark 2.3
  • Remark 2.4
  • Definition 2.4
  • Definition 2.5
  • ...and 24 more