Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Novel Approach to Peng's Maximum Principle for McKean-Vlasov Stochastic Differential Equations

Johan Benedikt Spille, Wilhelm Stannat

TL;DR

This work addresses Peng's maximum principle for McKean-Vlasov SDEs, where mean-field interactions via the law $\mu_t$ complicate second-order dualization. It introduces a third adjoint, a conditional MV-BSDE on an extended product space with two independent copies, to dualize the quadratic mixed Lions derivatives and to preserve a consistent proof framework. The main result yields a Peng-type necessary condition expressed with a second-level adjoint term, while clarifying the role of the third adjoint as a methodological device rather than a term in the final principle. The proposed approach facilitates potential extensions to common-noise and infinite-dimensional settings, and provides a robust variational scheme for mean-field control problems under nonconvex controls.

Abstract

We present a novel approach to the proof of Peng's maximum principle for McKean-Vlasov stochastic differential equations (SDE). The main step is the introduction of a third adjoint equation, a conditional McKean-Vlasov backward SDE, to accommodate the dualization of quadratic terms containing two independent copies of the first-order variational process. This is an intrinsic extension of the maximum principle from Peng for standard SDE and gives a conceptually consistent proof. Our approach will be useful in further extensions to the common noise setting and the infinite dimensional setting.

A Novel Approach to Peng's Maximum Principle for McKean-Vlasov Stochastic Differential Equations

TL;DR

This work addresses Peng's maximum principle for McKean-Vlasov SDEs, where mean-field interactions via the law complicate second-order dualization. It introduces a third adjoint, a conditional MV-BSDE on an extended product space with two independent copies, to dualize the quadratic mixed Lions derivatives and to preserve a consistent proof framework. The main result yields a Peng-type necessary condition expressed with a second-level adjoint term, while clarifying the role of the third adjoint as a methodological device rather than a term in the final principle. The proposed approach facilitates potential extensions to common-noise and infinite-dimensional settings, and provides a robust variational scheme for mean-field control problems under nonconvex controls.

Abstract

We present a novel approach to the proof of Peng's maximum principle for McKean-Vlasov stochastic differential equations (SDE). The main step is the introduction of a third adjoint equation, a conditional McKean-Vlasov backward SDE, to accommodate the dualization of quadratic terms containing two independent copies of the first-order variational process. This is an intrinsic extension of the maximum principle from Peng for standard SDE and gives a conceptually consistent proof. Our approach will be useful in further extensions to the common noise setting and the infinite dimensional setting.
Paper Structure (7 sections, 6 theorems, 48 equations)

This paper contains 7 sections, 6 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Variational Equations
  4. The Adjoint Equations
  5. The Lifting and the Second-Level Second-Order Adjoint Equation
  6. The Duality Relations
  7. The Expansion of the Cost Functional and the Maximum Principle

Key Result

Lemma 2.3

Let $\varphi \in C_b^{2,1}(\mathcal{P}_2(\mathbb{R}^d),\mathbb{R})$. Then, for any given $\vartheta_0 \in L^2(\Omega, \mathbb{R}^d)$ we have the following second-order expansion, for all $\vartheta\in L^2(\Omega, \mathbb{R}^d)$ where $\eta:=\vartheta-\vartheta_0$, and for all $\vartheta \in L^2(\Omega, \mathbb{R}^d)$ the remainder $R(\mathcal{L}(\vartheta), \mathcal{L}(\vartheta_0))$ satisfies the

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: BuckdahnPeng2017 Lemma 2.1
  • Remark 2.4
  • Definition 2.5
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 11 more