# Bellman equation and viscosity solutions for mean-field stochastic control problem

arXiv: https://arxiv.org/abs/1512.07866 · HTML: https://mlsanity.com/paper/1512.07866/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Dec 2015.

## Built on by (most cited first, 11 of 11)

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- [A Survey on Large-Population Systems and Scalable Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2209.03859/) (2022)
- [Discrete-Time Mean Field Control with Environment States](https://mlsanity.com/paper/2104.14900/) (2021)
- [Learning Decentralized Partially Observable Mean Field Control for Artificial Collective Behavior](https://mlsanity.com/paper/2307.06175/) (2023)
- [Major-Minor Mean Field Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2303.10665/) (2023)
- [Policy Gradient for Continuous-Time Mean-Field Control](https://mlsanity.com/paper/2605.20718/) (2026)
- [Hypergraphon Mean Field Games](https://mlsanity.com/paper/2203.16223/) (2022)
- [Convergence of the Deep Galerkin Method for Finite State Mean Field Control Problems](https://mlsanity.com/paper/2405.13346/) (2024)
- [Nonlocal Mean Field Schr"odinger Bridge with Learned Interactions](https://mlsanity.com/paper/2606.04265/) (2026)
- [Actor-Critic Learning for Extended Mean Field Control with Deterministic Policies](https://mlsanity.com/paper/2607.11005/) (2026)
- [Sparse Mean Field Load Balancing in Large Localized Queueing Systems](https://mlsanity.com/paper/2312.12973/) (2023)

## Abstract (arXiv)

We consider the stochastic optimal control problem of McKean-Vlasov stochastic differential equation where the coefficients may depend upon the joint law of the state and control. By using feedback controls, we reformulate the problem into a deterministic control problem with only the marginal distribution of the process as controlled state variable, and prove that dynamic programming principle holds in its general form. Then, by relying on the notion of differentiability with respect to pro-bability measures recently introduced by P.L. Lions in [32], and a special Ito formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a veri-fication theorem in our McKean-Vlasov framework. We give explicit solutions to the Bellman equation for the linear quadratic mean-field control problem, with applications to the mean-variance portfolio selection and a systemic risk model. We also consider a notion of lifted visc-sity solutions for the Bellman equation, and show the viscosity property and uniqueness of the value function to the McKean-Vlasov control problem. Finally, we consider the case of McKean-Vlasov control problem with open-loop controls and discuss the associated dynamic programming equation that we compare with the case of closed-loop controls.