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Conditional McKean-Vlasov control

René Carmona, Ludovic Tangpi, Kaiwen Zhang

TL;DR

This work develops a comprehensive weak-formulation framework for conditional McKean–Vlasov control with hard killing, where the interaction occurs through the law conditioned on not exiting a domain. It establishes a general Pontryagin maximum principle tied to a generalized McKean–Vlasov BSDE, proves existence and uniqueness of optimal controls under convexity, and shows that optimal controls belong to a BMO class without requiring boundedness of the drift or controls. The paper provides two concrete applications: a Schrödinger-type problem with killing and a clean route from MKV control to potential mean field games, illustrating both penalized and equilibrium perspectives. The results advance the analysis and computation of interacting particle systems under conditioning, with implications for systemic risk modeling and the design of robust control strategies in stochastic settings.

Abstract

Conditional McKean-Vlasov control problems involve controlling McKean-Vlasov diffusions where the interaction occurs through the law of the state process conditionally on it staying in a domain. Introduced by Lions in his 2016 lectures at the Collège de France, these problems have notable applications, particularly in systemic risk. We establish well-posedness and provide a general characterization of optimal controls using a new Pontryagin maximum principle in the probabilistic weak formulation. Unlike the classical approach based on forward-backward systems, our results connect the control problem to a generalized McKean-Vlasov backward stochastic differential equation (BSDE). We illustrate our framework with two applications: a version of the Schrödinger problem with killing, and a construction of equilibria in potential mean field games via McKean-Vlasov control.

Conditional McKean-Vlasov control

TL;DR

This work develops a comprehensive weak-formulation framework for conditional McKean–Vlasov control with hard killing, where the interaction occurs through the law conditioned on not exiting a domain. It establishes a general Pontryagin maximum principle tied to a generalized McKean–Vlasov BSDE, proves existence and uniqueness of optimal controls under convexity, and shows that optimal controls belong to a BMO class without requiring boundedness of the drift or controls. The paper provides two concrete applications: a Schrödinger-type problem with killing and a clean route from MKV control to potential mean field games, illustrating both penalized and equilibrium perspectives. The results advance the analysis and computation of interacting particle systems under conditioning, with implications for systemic risk modeling and the design of robust control strategies in stochastic settings.

Abstract

Conditional McKean-Vlasov control problems involve controlling McKean-Vlasov diffusions where the interaction occurs through the law of the state process conditionally on it staying in a domain. Introduced by Lions in his 2016 lectures at the Collège de France, these problems have notable applications, particularly in systemic risk. We establish well-posedness and provide a general characterization of optimal controls using a new Pontryagin maximum principle in the probabilistic weak formulation. Unlike the classical approach based on forward-backward systems, our results connect the control problem to a generalized McKean-Vlasov backward stochastic differential equation (BSDE). We illustrate our framework with two applications: a version of the Schrödinger problem with killing, and a construction of equilibria in potential mean field games via McKean-Vlasov control.

Paper Structure

This paper contains 36 sections, 20 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Main contributions
  3. The role of the weak formulation
  4. Related work
  5. Preliminaries and main results
  6. Notation
  7. Problem formulation
  8. The Pontryagin maximum principle in the weak formulation
  9. The necessary condition
  10. The sufficient condition
  11. Wellposedness results
  12. Applications to Schrödinger-type problems and mean field games
  13. Potential mean field games
  14. Discussion and examples
  15. The Le Cam distance
  16. ...and 21 more sections

Key Result

Proposition 2.3

Under Assumption asmp: beta, for any $\alpha \in \mathbb{A}_{BMO}$, there exists a unique $\mathbb{P}^\alpha$ equivalent to $\mathbb{P}$ satisfying eq: p def. In particular, $\mathbb{A}_{BMO} \subset \mathbb{A}$. Moreover, for any $\alpha \in \mathbb{A}$, there is at most one $\mathbb{P}^\alpha \in

Theorems & Definitions (57)

  • Definition 2.1
  • Proposition 2.3
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.10
  • Remark 2.11
  • Theorem 2.12
  • Remark 2.13
  • ...and 47 more