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Itô-Wentzell formulas for semimartingale conditional laws with applications to mean-field control

Nizar Touzi, Mehdi Talbi

TL;DR

The paper extends Itô and Itô-Wentzell calculus to flows of conditional laws driven by general càdlàg semimartingales, enabling discontinuous law dynamics via a linear functional derivative and partition-based Taylor expansions. It derives a complete Itô decomposition for $u(m_t)$ and a random-field Itô-Wentzell formula for $U_t(m_t)$, incorporating jump corrections through cross-derivatives of the measure argument. These results are then applied to mean-field control problems with Poisson-type common noise and mean-field stopping problems with common noise, yielding HJB and obstacle equations on Wasserstein space and clarifying the role of jump terms in the dynamic programming framework. The framework unifies continuous and jump-type common noises within mean-field control and stopping, extending prior work and providing a robust toolkit for McKean-Vlasov-type control problems.

Abstract

The present paper is an extension of Fadle-Touzi (2024). Following the same methodology, merely based on Taylor expansions, we establish the Itô and Itô-Wentzell formulae for flows of conditional distributions of general semimartingales, thus allowing for discontinuous semimartingales with possibly discontinuous flows of conditional marginals. We apply these results to derive the dynamic programming equations corresponding to mean field control problems with Poisson type common noise and mean field stopping problems with common noise.

Itô-Wentzell formulas for semimartingale conditional laws with applications to mean-field control

TL;DR

The paper extends Itô and Itô-Wentzell calculus to flows of conditional laws driven by general càdlàg semimartingales, enabling discontinuous law dynamics via a linear functional derivative and partition-based Taylor expansions. It derives a complete Itô decomposition for and a random-field Itô-Wentzell formula for , incorporating jump corrections through cross-derivatives of the measure argument. These results are then applied to mean-field control problems with Poisson-type common noise and mean-field stopping problems with common noise, yielding HJB and obstacle equations on Wasserstein space and clarifying the role of jump terms in the dynamic programming framework. The framework unifies continuous and jump-type common noises within mean-field control and stopping, extending prior work and providing a robust toolkit for McKean-Vlasov-type control problems.

Abstract

The present paper is an extension of Fadle-Touzi (2024). Following the same methodology, merely based on Taylor expansions, we establish the Itô and Itô-Wentzell formulae for flows of conditional distributions of general semimartingales, thus allowing for discontinuous semimartingales with possibly discontinuous flows of conditional marginals. We apply these results to derive the dynamic programming equations corresponding to mean field control problems with Poisson type common noise and mean field stopping problems with common noise.

Paper Structure

This paper contains 9 sections, 6 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Itô's formula
  3. Itô-Wentzell's formula
  4. Applications to stochastic control
  5. Mean field control with Poisson type common noise
  6. Mean field optimal stopping with common noise
  7. Technical Results
  8. Proof of Lemma \ref{['lem:cv-bracket']}
  9. Proof of Lemma \ref{['lem:cv2']}

Key Result

Theorem 2.2

Let $X \!:=\! V \!+\! M$ be a càdlàg semimartingale with finite variation process $V$ and local martingale $M$ satisfying: Given a sub-sigma algebra ${\cal F}^0 \subset {\cal F}$, denote by $m_t$ the conditional law of $X_t$ given ${\cal F}^0$, $\mathbb{E}^0[ \cdot ] := \mathbb{E}^0[ \cdot | {\cal F}_T^0]$ and $\hat{\mathbb{E}}^0[ \cdot ] := \mathbb{E}^0[ \cdot | {\cal F}_T^0, X]$. Let $u \in C_b

Theorems & Definitions (10)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Example 2.4
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Proposition 4.1
  • Proposition 4.2