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Itō and Itō-Wentzell chain rule for flows of conditional laws of continuous semimartingales: an easy approach

Assil Fadle, Mehdi Talbi, Nizar Touzi

TL;DR

This work derives a general Itô–Wentzell formula for a random field on the Wasserstein space evaluated along the flow of conditional laws $m_t$ of a continuous semimartingale, using standard Itô calculus to avoid empirical-measure approximations. It first establishes an Itô formula for flows of conditional laws and then extends to a full Itô–Wentzell formula for random fields $U_t(m)$ with dynamics driven by finite-variation and martingale components. The results are illustrated through Brownian and semimartingale-factor examples, and an application to mean-field control with common noise yields a dynamic-programming/HJB equation involving Lions derivatives and cross-variations. The approach provides a simple, flexible framework for analyzing mean-field problems under common noise and partial information, with explicit decomposition formulas that facilitate further applications in stochastic control and game theory.

Abstract

We provide a general Itō\,-Wentzell formula for a random field of maps on the Wasserstein space of probability measures, defined by continuous semimartingales, and evaluated along the flow of conditional distributions of another continuous semimartingale. Our method follows standard arguments of Itō calculus, and thus bypasses the approximation by empirical measures commonly used in the existing literature. As an application, we derive the dynamic programming equation for a mean field stochastic control problem with common noise.

Itō and Itō-Wentzell chain rule for flows of conditional laws of continuous semimartingales: an easy approach

TL;DR

This work derives a general Itô–Wentzell formula for a random field on the Wasserstein space evaluated along the flow of conditional laws of a continuous semimartingale, using standard Itô calculus to avoid empirical-measure approximations. It first establishes an Itô formula for flows of conditional laws and then extends to a full Itô–Wentzell formula for random fields with dynamics driven by finite-variation and martingale components. The results are illustrated through Brownian and semimartingale-factor examples, and an application to mean-field control with common noise yields a dynamic-programming/HJB equation involving Lions derivatives and cross-variations. The approach provides a simple, flexible framework for analyzing mean-field problems under common noise and partial information, with explicit decomposition formulas that facilitate further applications in stochastic control and game theory.

Abstract

We provide a general Itō\,-Wentzell formula for a random field of maps on the Wasserstein space of probability measures, defined by continuous semimartingales, and evaluated along the flow of conditional distributions of another continuous semimartingale. Our method follows standard arguments of Itō calculus, and thus bypasses the approximation by empirical measures commonly used in the existing literature. As an application, we derive the dynamic programming equation for a mean field stochastic control problem with common noise.
Paper Structure (9 sections, 6 theorems, 38 equations)

This paper contains 9 sections, 6 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Itō's formula
  4. Itō -Wentzell's formula
  5. Examples
  6. Brownian Case
  7. Semimartingale factor random field model
  8. Application to Mean-Field Control: HJB Equation
  9. Appendix: Proof of Lemma \ref{['lemma']}

Key Result

Lemma 2.1

Let $X$ be a semimartingale with decomposition $X=A+M$ into a finite variation process $A$ and a martingale $M$ satisfying $\mathbb{E}\left[ |A|_{\rm{TV}}^2 + \langle M \rangle_T^2 \right] < \infty$. Let $(H^n)_{n\ge 0}$ be a simple approximation of a matrix-valued bounded progressively measurable $

Theorems & Definitions (6)

  • Lemma 2.1
  • Theorem 3.2
  • Theorem 4.2
  • Corollary 4.3
  • Corollary 4.4
  • Proposition 5.1