# Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics

arXiv: https://arxiv.org/abs/1604.04057 · HTML: https://mlsanity.com/paper/1604.04057/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2016.

## Built on by (most cited first, 12 of 12)

- [Actor-Critic learning for mean-field control in continuous time](https://mlsanity.com/paper/2303.06993/) (2023)
- [Continuous-time q-learning for mean-field control with common noise, part-I: Theoretical foundations](https://mlsanity.com/paper/2604.27372/) (2026)
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- [Policy Gradient for Continuous-Time Mean-Field Control](https://mlsanity.com/paper/2605.20718/) (2026)
- [An Optimal Control Approach To Transformer Training](https://mlsanity.com/paper/2603.09571/) (2026)
- [Convergence of the Deep Galerkin Method for Finite State Mean Field Control Problems](https://mlsanity.com/paper/2405.13346/) (2024)
- [A Subgame Perfect Equilibrium Reinforcement Learning Approach to Time-inconsistent Problems](https://mlsanity.com/paper/2110.14295/) (2021)
- [Learning to Stop: Deep Learning for Mean Field Optimal Stopping](https://mlsanity.com/paper/2410.08850/) (2024)
- [Mean-Field PhiBE: Continuous-Time Mean-Field Reinforcement Learning from Discrete-Time Data](https://mlsanity.com/paper/2606.26498/) (2026)
- [Convergence analysis of controlled particle systems arising in deep learning: from finite to infinite sample size](https://mlsanity.com/paper/2404.05185/) (2024)
- [Learning Generative Dynamics with Soft Law Constraints: A McKean-Vlasov FBSDE Approach](https://mlsanity.com/paper/2605.08928/) (2026)
- [Robust Q-learning for mean-field control under Wasserstein uncertainty in common noise](https://mlsanity.com/paper/2606.20356/) (2026)

## Abstract (arXiv)

We study the optimal control of general stochastic McKean-Vlasov equation. Such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles (players) in mean-field interaction under common noise. Our first main result is to state a dynamic programming principle for the value function in the Wasserstein space of probability measures, which is proved from a flow property of the conditional law of the controlled state process. Next, by relying on the notion of differentiability with respect to probability measures due to P.L. Lions [32], and Ito's formula along a flow of conditional measures, we derive the dynamic programming Hamilton-Jacobi-Bellman equation, and prove the viscosity property together with a uniqueness result for the value function. Finally, we solve explicitly the linear-quadratic stochastic McKean-Vlasov control problem and give an application to an interbank systemic risk model with common noise.